Asymptotic behavior of plate equations driven by colored noise on unbounded domains

Abstract

This paper investigates mainly the asymptotic behavior of the nonautonomous random dynamical systems generated by the plate equations driven by colored noise defined on $\mathbb{R}^{n}$. First, we prove the well-posedness of the equation in the natural energy space. Secondly, we define a continuous cocycle associated with the solution operator. Finally, we establish the existence and uniqueness of random attractors of the equation by the uniform tail-ends estimates methods and the splitting technique.

1 Introduction

Colored noise was first introduced in [17, 21] in order to obtain information on the velocity of randomly moving particles, which cannot be obtained from white noise since the Wiener process is nowhere differentiable. Moreover, for many physical systems, the stochastic fluctuations are correlated and should be modeled by colored noise rather than white noise, see [14].

This paper is concerned with the asymptotic behavior of the plate equation driven by nonlinear colored noise in unbounded domains:

$$ \textstyle\begin{cases} u_{tt}+\alpha u_{t}+\Delta ^{2}u+\nu u +f(x,u)=g(x,t)+h(t,x,u)\zeta _{ \delta}(\theta _{t}\omega ),\quad t>\tau , x\in \mathbb{R}^{n}, \\ u(x,\tau )=u_{0}(x), \qquad u_{t}(x,\tau )=u_{1,0}(x),\quad x\in \mathbb{R}^{n}, \end{cases} $$

(1.1)

where $\tau \in \mathbb{R}$, α, ν are positive constants, f and h are given nonlinearity, $g \in L^{2}_{\mathrm{loc}}(\mathbb{R}, H^{1}(\mathbb{R}^{n}))$, and $\zeta _{\delta}$ is a colored noise with correlation time $\delta >0$.

The existence and uniqueness of pathwise random attractors of stochastic plate equations have been studied in [12, 13, 15, 16] in the case of bounded domains; and in [30–35] in the case of unbounded domains. We also mention that the global attractors of deterministic plate equations have been investigated in [2, 7, 9, 10, 24, 26–29, 37] in bounded domains, and in [5, 6, 11, 25, 36] on unbounded domains.

In all these publications ([30–35]), only the additive white noise and linear multiplicative white noise were considered. Note that the random equation (1.1) is driven by colored noise rather than white noise. In general, it is very difficult to study the asymptotic dynamics of differential equations driven by nonlinear white noise, including the random attractors. Indeed, only when the white noise is linear can the stochastic equations be transformed into a deterministic equations, then one can obtain the existence of random attractors of the plate equation (1.1). However, this transformation does not apply to stochastic equations driven by nonlinear white noise, and that is why we are currently unable to prove the existence of random attractors for systems with nonlinear white noise.

For the colored noise, even if it is nonlinear, we are able to show system (1.1) has a random attractor in $H^{2}(\mathbb{R}^{n})\times L^{2}(\mathbb{R}^{n})$, which is quite different from the nonlinear white noise. The reader is referred to [3, 4, 22, 23] for more details on random attractors of differential equations driven by colored noise. In this paper, instead of using white noise, we will consider the random equation (1.1) driven by nonlinear colored noise. The main aim of this paper is to obtain the existence and uniqueness of random attractors for (1.1) when the diffusion term h is a nonlinear continuous function.

Note that system (1.1) is defined in the unbounded domain $\mathbb{R}^{n}$ where the noncompactness of Sobolev embeddings on unbounded domains gives rise to difficulty in showing the pullback asymptotic compactness of solutions; to overcome this we use the tail-estimates method (as in [18]) and the splitting technique to obtain the pullback asymptotic compactness.

The rest of this article consists of four sections. In the next section, we define some functions sets and recall some useful results. In Sect. 3, we first establish the existence, uniqueness, and continuity of solutions in initial data of (1.1) in $H^{2}(\mathbb{R}^{n})\times L^{2}(\mathbb{R}^{n})$, then define a nonautonomous random dynamical system based on the solution operator of problem (1.1). The last two sections are devoted to deriving necessary estimates of solutions of (1.1) and the existence of random attractors.

Throughout the paper, the inner product and the norm of $L^{2}(\mathbb{R}^{n})$ will be denoted by $(\cdot , \cdot )$ and $\|\cdot \|$, respectively. The letters c and $c_{i}$ ($i = 1, 2, \ldots$) are generic positive constants that may depend on some parameters in the contexts.

2 Asymptotic compactness of cocycles

In this section, we define some functions sets and recall some useful results, see [19, 20]. These results will be used to establish the asymptotic compactness of the solutions and attractor for the random plate equation defined on the entire space $\mathbb{R}^{n}$.

From now on, we assume $(\Omega ,\mathcal{F},P)$ is the canonical probability space where $\Omega =\{\omega \in C(\mathbb{R},\mathbb{R}):\omega (0)=0\}$ with compact-open topology, $\mathcal{F}$ is the Borel σ-algebra of Ω, and P is the Wiener measure on $(\Omega ,\mathcal{F})$. Recall the standard group of transformations $\{\theta _{t}\}_{t\in \mathbb{R}}$ on Ω:

$$ \theta _{t}\omega (\cdot )=\omega (t+\cdot )-\omega (t),\quad \forall t \in \mathbb{R} \text{ and } \forall \omega \in \Omega . $$

Let X be a Banach space with norm $\|\cdot \|_{X}$. Suppose $\Phi :\mathbb{R}^{+}\times \mathbb{R}\times \Omega \times X \rightarrow X$ is a continuous cocycle on X over $(\Omega ,\mathcal{F},P,\{\theta _{t}\}_{t\in \mathbb{R}})$. Let $\mathcal{D}$ be a collection of some families of the nonempty subset of X:

$$ \mathcal{D}=\bigl\{ D=\bigl\{ D(\tau ,\omega )\subseteq X:D(\tau ,\omega )\neq \emptyset ,\tau \in \mathbb{R},\omega \in \Omega \bigr\} \bigr\} . $$

Suppose Φ has a $\mathcal{D}$-pullback absorbing set $K=\{K(\tau ,\omega ):\tau \in \mathbb{R},\omega \in \Omega \}\in \mathcal{D}$; that is, for every $\tau \in \mathbb{R}$, $\omega \in \Omega$, and $D\in \mathcal{D}$ there exists $T=T(\tau ,\omega ,D)>0$ such that for all $t\geq T$,

$$ \Phi \bigl(t,\tau -t,\theta _{-t}\omega ,D(\tau -t,\theta _{-t}\omega )\bigr) \subseteq K(\tau ,\omega ). $$

(2.1)

Assume that

$$ \Phi (t,\tau ,\omega ,x)=\Phi _{1}(t,\tau ,\omega ,x)+\Phi _{2}(t, \tau ,\omega ,x), \quad \forall t\in \mathbb{R}^{+}, \tau \in \mathbb{R}, \omega \in \Omega , x\in X, $$

(2.2)

where both $\Phi _{1}$ and $\Phi _{2}$ are mappings from $\mathbb{R}^{+}\times \mathbb{R}\times \Omega \times X$ to X.

Given $k\in \mathbb{N}$, denote by $\mathcal{O}_{k}=\{x\in \mathbb{R}^{n}:|x|< k\}$ and $\tilde{\mathcal{O}}_{k}=\{x\in \mathbb{R}^{n}:|x|>k\}$. Let X be a Banach space with norm $\|\cdot \|_{X}$ that consists of some functions defined on $\mathbb{R}^{n}$. Given a function $u:\mathbb{R}^{n}\rightarrow \mathbb{R}$, the restrictions of u to $\mathcal{O}_{k}$ and $\tilde{\mathcal{O}}_{k}$ are written as $u|_{\mathcal{O}_{k}}$ and $u|_{\tilde{\mathcal{O}}_{k}}$, respectively. Denote by

$$ X_{\mathcal{O}_{k}}=\{u|_{\mathcal{O}_{k}}:u\in X\}\quad \text{and}\quad X_{ \tilde{\mathcal{O}}_{k}}= \{u|_{\tilde{\mathcal{O}}_{k}}:u\in X\}. $$

Suppose $X_{\mathcal{O}_{k}}$ and $X_{\tilde{\mathcal{O}}_{k}}$ are Banach spaces with norm $\|\cdot \|_{\mathcal{O}_{k}}$ and $\|\cdot \|_{\tilde{\mathcal{O}}_{k}}$, respectively, and

$$ \Vert u \Vert _{X}\leq \Vert u|_{\mathcal{O}_{k}} \Vert _{\mathcal{O}_{k}}+ \Vert u|_{ \tilde{\mathcal{O}}_{k}} \Vert _{\tilde{\mathcal{O}}_{k}}, \quad \forall u \in X. $$

(2.3)

We further assume that for every $\delta >0$, $\tau \in \mathbb{R}$, and $\omega \in \Omega$, there exists $t_{0}=t_{0}(\delta ,\tau ,\omega ,K)>0$ and $k_{0}=k_{0}(\delta ,\tau ,\omega )\geq 1$ such that

$$ \bigl\Vert \Phi (t_{0},\tau -t_{0},\theta _{-t_{0}}\omega ,x)|_{ \tilde{\mathcal{O}}_{k_{0}}} \bigr\Vert _{\tilde{\mathcal{O}}_{k_{0}}}< \delta , \quad \forall x\in K(\tau -t_{0},\theta _{-t_{0}}\omega ), $$

(2.4)

and

$$\begin{aligned}& \Phi _{1}\bigl(t_{0},\tau -t_{0},\theta _{-t_{0}}\omega ,K(\tau -t_{0}, \theta _{-t_{0}}\omega ) \bigr)|_{\mathcal{O}_{k_{0}}} \\& \quad \text{has a finite cover of balls of radius } \delta \text{ in } X|_{ \mathcal{O}_{k_{0}}}. \end{aligned}$$

(2.5)

In addition, we assume that for every $k\in \mathbb{N}$, $t\in \mathbb{R}^{+}$, $\tau \in \mathbb{R}$, and $\omega \in \Omega$, the set

$$ \Phi _{2}\bigl(t,\tau -t,\theta _{-t}\omega ,K(\tau -t, \theta _{-t}\omega )\bigr)\quad \text{is precompact in } X|_{\mathcal{O}_{k}}. $$

(2.6)

Theorem 2.1

If (2.1)–(2.6) hold, then the cocycle Φ is $\mathcal{D}$-pullback asymptotically compact in X; that is, the sequence $\{ \Phi (t_{n},\tau -t_{n},\theta _{-t_{n}}\omega ,x_{n})\}^{\infty}_{n=1}$ is precompact in X for any $\tau \in \mathbb{R}$, $\omega \in \Omega$, $D\in \mathcal{D}$, $t_{n} \rightarrow \infty$ monotonically, and $x_{n}\in D(\tau -t_{n}, \theta _{-t_{n}}\omega )$.

Theorem 2.2

Let $\mathcal{D}$ be an inclusion closed collection of some families of nonempty bounded subsets of X, and Φ be a continuous cocycle on X over $(\Omega ,\mathcal{F}, P ,\{\theta _{t}\}_{t\in \mathbb{R} })$. Then, Φ has a unique $\mathcal{D}$-pullback random attractor $\mathcal{A}$ in $\mathcal{D}$ if Φ is $\mathcal{D}$-pullback asymptotically compact in X and Φ has a closed measurable $\mathcal{D}$-pullback absorbing set K in $\mathcal{D}$.

3 Cocycles of random plate equations

In this section, we first establish the existence of a solution for problem (1.1), then we define a nonautonomous cocycle of (1.1).

Given $\delta >0$, let $\zeta _{\delta}(\theta _{t}\omega )$ be the unique stationary solution of the stochastic equation:

$$ d\zeta _{\delta}+\frac{1}{\delta}\zeta _{\delta }\,dt= \frac{1}{\delta}\,dW, $$

(3.1)

where W is a two-sided real-valued Wiener process on $(\Omega ,\mathcal{F},P)$. The process $\zeta _{\delta}(\theta _{t}\omega )$ is called one-dimensional colored noise. Recall that there exists a $\theta _{t}$-invariant subset of full measure (see [1]), which is still denoted by Ω, such that for all $\omega \in \Omega$, $\zeta _{\delta}(\theta _{t}\omega )$ is continuous in $t\in \mathbb{R}$ and

$$ \lim_{t\rightarrow \pm \infty} \frac{ \vert \zeta _{\delta}(\theta _{t}\omega ) \vert }{t}=0, \quad \text{for } 0< \delta \leq 1. $$

Let −Δ denote the Laplace operator in $\mathbb{R}^{n}$, $A=\Delta ^{2}$ with the domain $D(A)=H^{4}(\mathbb{R}^{n})$. We can also define the powers $A^{\nu}$ of A for $\nu \in \mathbb{R}$. The space $V_{\nu} = D(A^{\frac{\nu}{4}})$ is a Hilbert space with the following inner product and norm

$$ (u,v)_{\nu}=\bigl(A^{\frac{\nu}{4}}u,A^{\frac{\nu}{4}}v\bigr), \qquad \Vert \cdot \Vert _{\nu}= \bigl\Vert A^{\frac{\nu}{4}}\cdot \bigr\Vert . $$

We introduce the following hypotheses to complete the uniform estimates.

Let $f:\mathbb{R}^{n}\times \mathbb{R}\rightarrow \mathbb{R}$ be a continuous function and $F(x,r)=\int ^{r}_{0}f(x,s)\,ds$ for all $x\in \mathbb{R}^{n}$, $r\in \mathbb{R}$ and $s, s_{1}, s_{2}\in \mathbb{R}$,

$$\begin{aligned} &\liminf_{ \vert s \vert \rightarrow \infty}\inf_{x\in \mathbb{R}^{n}}\bigl(f(x,s)s \bigr)>0, \end{aligned}$$

(3.2)

$$\begin{aligned} &f(x,0)=0,\qquad \bigl\vert f(x,s_{1})-f(x,s_{2}) \bigr\vert \leq \alpha _{1}\bigl(\varphi (x)+ \vert s_{1} \vert ^{p}+ \vert s_{2} \vert ^{p}\bigr) \vert s_{1}-s_{2} \vert , \end{aligned}$$

(3.3)

$$\begin{aligned} &F(x,s)+\varphi _{1}(x)\geq 0, \end{aligned}$$

(3.4)

where $p>0$ for $1\leq n\leq 4$ and $0< p\leq \frac{4}{n-4}$ for $n\geq 5$, $\alpha _{1}$ is a positive constant, $\varphi _{1}\in L^{1}(\mathbb{R}^{n})$, and $\varphi \in L^{\infty}(\mathbb{R}^{n})$.

Let $h:\mathbb{R}\times \mathbb{R}^{n}\times \mathbb{R}\rightarrow \times \mathbb{R}$ be continuous such that for all $t, s, s_{1}, s_{2}\in \mathbb{R}$ and $x\in \mathbb{R}^{n}$,

$$\begin{aligned} & \bigl\vert h(t,x,s) \bigr\vert \leq \alpha _{2} \vert s \vert +\varphi _{2}(t,x), \end{aligned}$$

(3.5)

$$\begin{aligned} & \bigl\vert h(t,x,s_{1})-h(t,x,s_{2}) \bigr\vert \leq \alpha _{3} \vert s_{1}-s_{2} \vert , \end{aligned}$$

(3.6)

where $\alpha _{2}$ and $\alpha _{3}$ are positive constants, and $\varphi _{2}\in L^{2}_{\mathrm{loc}}(\mathbb{R},L^{2}(\mathbb{R}^{n}))$.

Definition 3.1

Given $\tau \in \mathbb{R}$, $\omega \in \Omega$, $T>0$, $u_{0}\in H^{2}( \mathbb{R}^{n})$, and $u_{1,0}\in L^{2}(\mathbb{R}^{n})$, a function $u(\cdot ,\tau ,\omega ,u_{0},u_{1,0}):[\tau ,\tau +T]\rightarrow H^{2}( \mathbb{R}^{n})$ is called a (weak) solution of (1.1) if the following conditions are fulfilled:

(i) $u(\cdot ,\tau ,\omega ,u_{0},u_{1,0})\in L^{\infty}(\tau ,\tau +T;H^{2}( \mathbb{R}^{n}))\cap C([\tau ,\tau +T],L^{2}(\mathbb{R}^{n}))$ with $u(\tau ,\tau ,\omega ,u_{0},u_{1,0})=u_{0}$, $u_{t}(\cdot ,\tau , \omega ,u_{0},u_{1,0})\in L^{\infty}(\tau ,\tau +T;L^{2}(\mathbb{R}^{n})) \cap C([\tau ,\tau +T],L^{2}(\mathbb{R}^{n}))$ with $u_{t}(\tau ,\tau ,\omega ,u_{0},u_{1,0})=u_{1,0}$.

(ii) $u(t,\tau ,\cdot ,u_{0},u_{1,0}):\Omega \rightarrow H^{2}(\mathbb{R}^{n})$ is $(\mathcal{F},\mathcal{B}(H^{2}(\mathbb{R}^{n}))$-measurable, and $u_{t}(t,\tau ,\cdot ,u_{0},u_{1,0}):\Omega \rightarrow L^{2}( \mathbb{R}^{n})$ is $(\mathcal{F},\mathcal{B}(L^{2}(\mathbb{R}^{n}))$-measurable.

(iii) For all $\xi \in C^{\infty}_{0}((\tau ,\tau +T)\times \mathbb{R}^{n})$,

$$\begin{aligned} &- \int ^{\tau +T}_{\tau}(u_{t},\xi _{t})\,dt+\alpha \int ^{\tau +T}_{ \tau}(u_{t},\xi )\,dt+ \int ^{\tau +T}_{\tau}(\Delta u,\Delta \xi )\,dt \\ &\qquad {} +\nu \int ^{\tau +T}_{\tau}(u,\xi )\,dt+ \int ^{\tau +T}_{\tau} \int _{\mathbb{R}^{n}}f\bigl(x,u(t,x)\bigr)\xi (t,x)\,dx\,dt \\ &\quad = \int ^{\tau +T}_{\tau}\bigl(g(t,x),\xi \bigr)\,dt+ \int ^{\tau +T}_{\tau} \int _{ \mathbb{R}^{n}}h\bigl(t,x,u(t,x)\bigr) \zeta _{\delta}(\theta _{t}\omega )\xi (t,x)\,dx\,dt. \end{aligned}$$

In order to investigate the long-time dynamics, we are now ready to prove the existence and uniqueness of solutions of (1.1). We first recall the following well-known existence and uniqueness of solutions for the corresponding linear plate equations of (1.1).

Lemma 3.1

Let $u_{0}\in H^{2}(\mathbb{R}^{n})$, $u_{1,0}\in L^{2}(\mathbb{R}^{n})$ and $g\in L^{1}(\tau ,\tau +T;L^{2}(\mathbb{R}^{n}))$ with $\tau \in \mathbb{R}$ and $T>0$. Then, the linear plate equation

$$ u_{tt}+\alpha u_{t}+\Delta ^{2}u+\nu u=g(t), \quad \tau < t\leq \tau +T, $$

with the initial conditions

$$ u(\tau )=u_{0}, \quad \textit{and}\quad u_{t}(\tau )=u_{1,0}, $$

possesses a unique solution u in the sense of Definition 3.1. In addition,

$$ u\in C\bigl([\tau ,\tau +T],H^{2}\bigl(\mathbb{R}^{n}\bigr) \bigr)\quad \textit{and}\quad u_{t} \in C\bigl([\tau ,\tau +T],L^{2} \bigl(\mathbb{R}^{n}\bigr)\bigr), $$

and there exists a positive number C depending only on ν (but independent of τ, T, $u_{0}$, $u_{1,0}$, and g) such that for all $t \in [\tau , \tau + T]$,

$$ \bigl\Vert u(t) \bigr\Vert _{H^{2}(\mathbb{R}^{n})}+ \bigl\Vert u_{t}(t) \bigr\Vert \leq C\biggl( \Vert u_{0} \Vert _{H^{2}( \mathbb{R}^{n})}+ \Vert u_{1,0} \Vert + \int ^{\tau +T}_{\tau} \bigl\Vert g(t) \bigr\Vert \,dt \biggr). $$

(3.7)

Furthermore, the solution u satisfies the energy equation

$$ \frac{d}{dt}\bigl( \Vert u_{t} \Vert ^{2}+ \Vert \Delta u \Vert ^{2}+\nu \Vert u \Vert ^{2} \bigr)=-2\alpha \Vert u_{t} \Vert ^{2}+2 \bigl(g(t),u_{t}\bigr), $$

(3.8)

and

$$ \frac{d}{dt}\bigl(u(t),u_{t}(t)\bigr)+\alpha \bigl(u(t),u_{t}(t)\bigr)+ \bigl\Vert \Delta u(t) \bigr\Vert ^{2}+ \nu \bigl\Vert u(t) \bigr\Vert ^{2}= \bigl\Vert u_{t}(t) \bigr\Vert ^{2}+\bigl(g(t),u(t)\bigr), $$

(3.9)

for almost all $t \in [\tau , \tau + T]$.

Theorem 3.1

Let $\tau \in \mathbb{R}$, $u_{0}\in H^{2}(\mathbb{R}^{n})$, $u_{1,0}\in L^{2}( \mathbb{R}^{n})$. Suppose (3.2)–(3.6) hold, then:

(a) Problem (1.1) possesses a solution u in the sense of Definition 3.1;

(b) The solution u to problem (1.1) is unique, continuous with initial data in $H^{2}(\mathbb{R}^{n})\times L^{2}(\mathbb{R}^{n})$, and

$$ u\in C\bigl([\tau ,\tau +T], H^{2}\bigl(\mathbb{R}^{n}\bigr) \bigr) \quad \textit{and}\quad u_{t} \in C\bigl([\tau ,\tau +T], L^{2} \bigl(\mathbb{R}^{n}\bigr)\bigr). $$

(3.10)

Moreover, the solution u to problem (1.1) satisfies the energy equation:

$$\begin{aligned} &\frac{d}{dt}\biggl( \Vert u_{t} \Vert ^{2}+\nu \Vert u \Vert ^{2}+ \Vert \Delta u \Vert ^{2}+2 \int _{ \mathbb{R}^{n}}F\bigl(x,u(t,x)\bigr)\,dx\biggr)+2\alpha \Vert u_{t} \Vert ^{2} \\ &\quad =2\bigl(g(t),u_{t}\bigr)+2\zeta _{\delta}(\theta _{t}\omega ) \int _{\mathbb{R}^{n}}h\bigl(t,x,u(t,x)\bigr)u_{t}(t,x)\,dx \end{aligned}$$

(3.11)

for almost all $t\in [\tau ,\tau +T]$.

Proof

The proof will be divided into four steps. We first construct a sequence of approximate solutions, and then derive uniform estimates, in the last two steps we take the limit of those approximate solutions to prove the uniqueness of the solutions.

Step (i): Approximate solutions Given $k\in \mathbb{N}$, define a function $\eta _{k}:\mathbb{R}\rightarrow \mathbb{R}$ by

$$ \eta _{k}(s)=\textstyle\begin{cases} s, & \text{if } -k\leq s \leq k, \\ k, &\text{if } s>k, \\ -k, & \text{if } s< -k. \end{cases} $$

(3.12)

Then, for every fixed $k\in \mathbb{N}$, the function $\eta _{k}$ as defined by (3.12) is bounded and Lipschitz continuous; more precisely, for all $s, s_{1}, s_{2}\in \mathbb{R}$

$$ \eta _{k}(0)=0, \qquad \bigl\vert \eta _{k}(s) \bigr\vert \leq \vert s \vert \quad \text{and}\quad \bigl\vert \eta _{k}(s_{1})- \eta _{k}(s_{2}) \bigr\vert \leq \vert s_{1}-s_{2} \vert . $$

(3.13)

For all $x\in \mathbb{R}^{n}$ and $t, s\in \mathbb{R}$, denote

$$ \begin{aligned}&f_{k}(x,s)=f\bigl(x,\eta _{k}(s)\bigr),\qquad F_{k}(x,s)= \int ^{s}_{0}f_{k}(x,r)\,dr\quad \text{and} \\ &h_{k}(t,x,s)=h\bigl(t,x,\eta _{k}(s)\bigr). \end{aligned}$$

(3.14)

By (3.2) we know that there exists $k_{0}\in \mathbb{N}$ such that for all $|s|\geq k_{0}$ and $x\in \mathbb{R}^{n}$,

$$ f(x,s)s>0, $$

(3.15)

thus, for all $k\geq k_{0}$ and $x\in \mathbb{R}^{n}$,

$$ f_{k}(x,k)>0,\qquad f_{k}(x,-k)< 0. $$

(3.16)

By (3.3), (3.4), (3.13), (3.14), and (3.16) we know that for all $s, s_{1}, s_{2}\in \mathbb{R}$ and $x\in \mathbb{R}^{n}$,

$$ \bigl\vert f_{k}(x,s_{1})-f_{k}(x,s_{2}) \bigr\vert \leq \alpha _{1}\bigl(\varphi (x)+ \vert s_{1} \vert ^{p}+ \vert s_{2} \vert ^{p}\bigr) \vert s_{1}-s_{2} \vert , \quad \forall k\geq 1, $$

(3.17)

and

$$ F_{k}(x,s)+\varphi _{1}(x)\geq 0, \quad \forall k\geq k_{0}. $$

(3.18)

By (3.17) we obtain that for all $s\in \mathbb{N}$ and $x\in \mathbb{R}^{n}$,

$$ \bigl\vert F_{k}(x,s) \bigr\vert \leq \alpha _{1}\bigl( \varphi (x) \vert s \vert ^{2}+ \vert s \vert ^{p+2} \bigr), \quad \forall k\geq 1. $$

(3.19)

By (3.5), (3.6), (3.13), and (3.14) we obtain that for all $k\geq 1$, $t, s, s_{1}, s_{2}\in \mathbb{R}$ and $x\in \mathbb{R}^{n}$,

$$\begin{aligned} & \bigl\vert h_{k}(t,x,s) \bigr\vert \leq \alpha _{2} \vert s \vert +\varphi _{2}(t,x), \end{aligned}$$

(3.20)

$$\begin{aligned} & \bigl\vert h_{k}(t,x,s_{1})-h_{k}(t,x,s_{2}) \bigr\vert \leq \alpha _{3} \vert s_{1}-s_{2} \vert . \end{aligned}$$

(3.21)

By (3.3), (3.13), and (3.14), we find that for all $k\in \mathbb{N}$, $s, s_{1}, s_{2}\in \mathbb{N}$ and $x\in \mathbb{R}^{n}$,

$$\begin{aligned} & \bigl\vert f_{k}(x,s) \bigr\vert \leq \alpha _{1}k \bigl(\varphi (x)+k^{p}\bigr), \end{aligned}$$

(3.22)

$$\begin{aligned} & \bigl\vert f_{k}(x,s_{1})-f_{k}(x,s_{2}) \bigr\vert \leq \alpha _{1}\bigl(\varphi (x)+2k^{p}\bigr) \vert s_{1}-s_{2} \vert . \end{aligned}$$

(3.23)

For every $k\in \mathbb{N}$, consider the following approximate system for $u_{k}$:

$$ \textstyle\begin{cases} \frac{\partial ^{2}}{\partial t^{2}}u_{k}+\alpha \frac{\partial}{\partial t} u_{k}+\Delta ^{2}u_{k}+\nu u_{k} +f_{k}( \cdot ,u_{k})=g(\cdot ,t)+h_{k}(t,\cdot ,u_{k})\zeta _{\delta}( \theta _{t}\omega ),\quad t>\tau , \\ u_{k}(\tau )=u_{0}, \qquad \frac{\partial}{\partial t}u_{k}(\tau )=u_{1,0}. \end{cases} $$

(3.24)

From (3.21), (3.23), $\varphi \in L^{\infty}(\mathbb{R}^{n})$, and the standard method (see, e.g., [8]), it follows that for each $\tau \in \mathbb{R}$, $\omega \in \Omega$, $u_{0}\in H^{2}(\mathbb{R}^{n})$, $u_{1,0}\in L^{2}(\mathbb{R}^{n})$, problem (3.24) has a unique global solution $u_{k}$ defined on $[\tau ,\tau +T]$ for every $T>0$ in the sense of Definition 3.1. In particular, $u_{k}(\cdot ,\tau ,\omega ,u_{0})\in C([\tau ,\tau +T],H^{2}( \mathbb{R}^{n}))$ and $u_{k}(t,\tau ,\omega ,u_{0})$ is measurable with respect to $\omega \in \Omega$ in $H^{2}(\mathbb{R}^{n})$ for every $t\in [\tau ,\tau +T]$. Similarly, $\partial _{t}u_{k}(\cdot ,\tau ,\omega ,u_{0})\in C([\tau ,\tau +T],L^{2}( \mathbb{R}^{n}))$ and $\partial _{t}u_{k}(t,\tau ,\omega ,u_{0})$ is measurable with respect to $\omega \in \Omega$ in $L^{2}(\mathbb{R}^{n})$ for every $t\in [\tau ,\tau +T]$. Furthermore, the solution $u_{k}$ satisfies the energy equation:

$$\begin{aligned} &\frac{d}{dt}\biggl( \Vert \partial _{t}u_{k} \Vert ^{2}+\nu \Vert u_{k} \Vert ^{2}+ \Vert \Delta u_{k} \Vert ^{2}+2 \int _{\mathbb{R}^{n}}F_{k}\bigl(x,u_{k}(t,x)\bigr)\,dx \biggr)+2\alpha \Vert \partial _{t}u_{k} \Vert ^{2} \\ &\quad =2\bigl(g(t),\partial _{t}u_{k}\bigr)+2\zeta _{\delta}(\theta _{t}\omega ) \int _{\mathbb{R}^{n}}h_{k}\bigl(t,x,u_{k}(t,x)\bigr) \partial _{t}u_{k}(t,x)\,dx \end{aligned}$$

(3.25)

for almost all $t\in [\tau ,\tau +T]$. Next, we use the energy equation (3.25) to derive a uniform estimate on the sequence $\{u_{k}\}^{\infty}_{k=1}$.

Step (ii): Uniform estimates

For the last term on the right-hand side of (3.25), by (3.21) we have

$$\begin{aligned} &2\zeta _{\delta}(\theta _{t}\omega ) \int _{\mathbb{R}^{n}}h_{k}\bigl(t,x,u_{k}(t,x)\bigr) \partial _{t}u_{k}(t,x)\,dx \\ &\quad \leq2 \bigl\vert \zeta _{\delta}(\theta _{t}\omega ) \bigr\vert \biggl(\alpha _{2} \int _{ \mathbb{R}^{n}} \bigl\vert u_{k}(t,x) \bigr\vert \cdot \bigl\vert \partial _{t}u_{k}(t,x) \bigr\vert \,dx + \int _{ \mathbb{R}^{n}} \bigl\vert \varphi _{2}(t,x) \bigr\vert \cdot \bigl\vert \partial _{t}u_{k}(t,x) \bigr\vert \,dx \biggr) \\ &\quad \leq \bigl\vert \zeta _{\delta}(\theta _{t}\omega ) \bigr\vert \bigl(\alpha _{2} \bigl\Vert u_{k}(t) \bigr\Vert ^{2}+(1+ \alpha _{2}) \bigl\Vert \partial _{t}u_{k}(t) \bigr\Vert ^{2}+ \bigl\Vert \varphi _{2}(t) \bigr\Vert ^{2}\bigr). \end{aligned}$$

(3.26)

By Young’s inequality, we obtain

$$\begin{aligned} 2\bigl(g(t),\partial _{t}u_{k}\bigr)\leq \bigl\Vert \partial _{t}u_{k}(t) \bigr\Vert ^{2}+ \bigl\Vert g(t) \bigr\Vert ^{2}. \end{aligned}$$

(3.27)

By (3.25)–(3.27), it follows that for almost all $t\in [\tau ,\tau +T]$,

$$\begin{aligned} &\frac{d}{dt}\biggl( \Vert \partial _{t}u_{k} \Vert ^{2}+\nu \Vert u_{k} \Vert ^{2}+ \Vert \Delta u_{k} \Vert ^{2}+2 \int _{\mathbb{R}^{n}}F_{k}\bigl(x,u_{k}(t,x)\bigr)\,dx \biggr)+2\alpha \Vert \partial _{t}u_{k} \Vert ^{2} \\ &\quad \leq c_{1}\bigl(1+ \bigl\vert \zeta _{\delta}(\theta _{t}\omega ) \bigr\vert \bigr) \bigl( \bigl\Vert u_{k}(t) \bigr\Vert ^{2}+ \bigl\Vert \partial _{t}u_{k}(t) \bigr\Vert ^{2}\bigr) + \bigl\vert \zeta _{\delta}(\theta _{t}\omega ) \bigr\vert \cdot \bigl\Vert \varphi _{2}(t) \bigr\Vert ^{2}+ \bigl\Vert g(t) \bigr\Vert ^{2}, \end{aligned}$$

(3.28)

where $c_{1} > 0$ depends only on $\alpha _{2}$, but independent of k.

By (3.18) and (3.28) we obtain

$$\begin{aligned} &\frac{d}{dt}\biggl( \Vert \partial _{t}u_{k} \Vert ^{2}+\nu \Vert u_{k} \Vert ^{2}+ \Vert \Delta u_{k} \Vert ^{2}+2 \int _{\mathbb{R}^{n}}F_{k}\bigl(x,u_{k}(t,x)\bigr)\,dx \biggr) \\ &\quad \leq c_{2}\bigl(1+ \bigl\vert \zeta _{\delta}(\theta _{t}\omega ) \bigr\vert \bigr) \biggl( \bigl\Vert \partial _{t}u_{k}(t) \bigr\Vert ^{2}+\nu \bigl\Vert u_{k}(t) \bigr\Vert ^{2}+ \Vert \Delta u_{k} \Vert ^{2}+2 \int _{\mathbb{R}^{n}}F_{k}\bigl(x,u_{k}(t,x)\bigr)\,dx \biggr) \\ &\qquad {}+ \bigl\vert \zeta _{\delta}(\theta _{t}\omega ) \bigr\vert \cdot \bigl\Vert \varphi _{2}(t) \bigr\Vert ^{2}+2c_{1}\bigl(1+ \bigl\vert \zeta _{\delta}(\theta _{t}\omega ) \bigr\vert \bigr) \Vert \varphi _{1} \Vert _{L^{1}( \mathbb{R}^{n})}+ \bigl\Vert g(t) \bigr\Vert ^{2}, \end{aligned}$$

(3.29)

where $c_{2} > 0$ depends only on ν and $\alpha _{2}$, but is independent of k.

Multiplying (3.29) by $e^{-c_{2}\int ^{t}_{0}(1+|\zeta _{\delta}(\theta _{r}\omega )|)\,dr}$, and then integrating the inequality on $(\tau , t)$, we have

$$\begin{aligned} & \Vert \partial _{t}u_{k} \Vert ^{2}+\nu \Vert u_{k} \Vert ^{2}+ \Vert \Delta u_{k} \Vert ^{2}+2 \int _{\mathbb{R}^{n}}F_{k}\bigl(x,u_{k}(t,x)\bigr)\,dx \\ &\quad \leq e^{c_{2}\int ^{t}_{\tau}(1+ \vert \zeta _{\delta}(\theta _{r}\omega ) \vert )\,dr} \biggl( \Vert u_{1,0} \Vert ^{2}+ \nu \Vert u_{0} \Vert ^{2}+ \Vert \Delta u_{0} \Vert ^{2}+2 \int _{ \mathbb{R}^{n}}F_{k}\bigl(x,u_{0}(x)\bigr)\,dx \biggr) \\ &\qquad {}+ \int ^{t}_{\tau }e^{c_{2}\int ^{t}_{s}(1+ \vert \zeta _{\delta}(\theta _{r} \omega ) \vert )\,dr}\bigl( \bigl\vert \zeta _{\delta}(\theta _{s}\omega ) \bigr\vert \cdot \bigl\Vert \varphi _{2}(s) \bigr\Vert ^{2} \\ &\qquad {} +2c_{1} \bigl(1+ \bigl\vert \zeta _{\delta}(\theta _{s}\omega ) \bigr\vert \bigr) \Vert \varphi _{1} \Vert _{L^{1}(\mathbb{R}^{n})}+ \bigl\Vert g(s) \bigr\Vert ^{2}\bigr)\,ds. \end{aligned}$$

(3.30)

By (3.19) we obtain, for all $k\geq 1$,

$$\begin{aligned} 2 \int _{\mathbb{R}^{n}} \bigl\vert F_{k}\bigl(x,u_{0}(x) \bigr) \bigr\vert \,dx&\leq 2\alpha _{1} \bigl( \Vert \varphi \Vert _{L^{\infty}(\mathbb{R}^{n})} \Vert u_{0} \Vert ^{2} + \Vert u_{0} \Vert ^{p+2}_{L^{p+2}( \mathbb{R}^{n})} \bigr) \\ & \leq 2\alpha _{1} \bigl( \Vert \varphi \Vert _{L^{\infty}(\mathbb{R}^{n})} \Vert u_{0} \Vert ^{2} + \Vert u_{0} \Vert ^{p+2}_{H^{2}(\mathbb{R}^{n})} \bigr). \end{aligned}$$

(3.31)

Equations (3.30) and (3.31) imply that there exists a positive constant $c_{3}=c_{3}(\tau ,T,\varphi ,\varphi _{1},\varphi _{2}, g,\omega , \delta ,\alpha _{1},\nu )$ (but independent of k, $u_{0}$, and $u_{1,0}$) such that for all $t\in [\tau ,\tau +T]$ and $k\geq 1$,

$$\begin{aligned} & \Vert \partial _{t}u_{k} \Vert ^{2}+\nu \Vert u_{k} \Vert ^{2}+ \Vert \Delta u_{k} \Vert ^{2}+2 \int _{\mathbb{R}^{n}}F_{k}\bigl(x,u_{k}(t,x)\bigr)\,dx \\ &\quad \leq c_{3}+c_{3}\bigl(1+ \Vert u_{1,0} \Vert ^{2}+ \Vert u_{0} \Vert ^{p+2}_{H^{2}(\mathbb{R}^{n})} \bigr), \end{aligned}$$

which along with (3.18) show that for all $t\in [\tau ,\tau +T]$ and $k\geq k_{0}$,

$$\begin{aligned} & \Vert \partial _{t}u_{k} \Vert ^{2}+\nu \Vert u_{k} \Vert ^{2}+ \Vert \Delta u_{k} \Vert ^{2}+2 \int _{\mathbb{R}^{n}}F_{k}\bigl(x,u_{k}(t,x)\bigr)\,dx \\ &\quad \leq c_{3}+2 \Vert \varphi _{1} \Vert _{L^{1}(\mathbb{R}^{n})}+c_{3}\bigl(1+ \Vert u_{1,0} \Vert ^{2}+ \Vert u_{0} \Vert ^{p+2}_{H^{2}(\mathbb{R}^{n})} \bigr), \end{aligned}$$

(3.32)

thus,

$$ \{u_{k}\}^{\infty}_{k=1} \quad \text{is bounded in } L^{\infty}\bigl( \tau ,\tau +T;H^{2}\bigl(\mathbb{R}^{n} \bigr)\bigr) $$

(3.33)

and

$$ \{\partial _{t} u_{k}\}^{\infty}_{k=1} \quad \text{is bounded in } L^{ \infty}\bigl(\tau ,\tau +T;L^{2}\bigl( \mathbb{R}^{n}\bigr)\bigr). $$

(3.34)

By (3.17), there exists a positive constant $c_{4}=c_{4}(p,n,\alpha _{1})$ such that

$$ \int _{\mathbb{R}^{n}} \bigl\vert f_{k}\bigl(x,u_{k}(t,x) \bigr) \bigr\vert ^{2}\,dx\leq c_{4} \biggl( \int _{\mathbb{R}^{n}} \bigl\vert \varphi (x) \bigr\vert ^{2}\,dx+ \int _{\mathbb{R}^{n}} \bigl\vert u_{k}(t,x) \bigr\vert ^{2(p+1)}\,dx \biggr), $$

which along with the embedding $H^{2}(\mathbb{R}^{n})\hookrightarrow L^{2(p+1)}(\mathbb{R}^{n})$ and the assumption $\varphi \in L^{\infty}(\mathbb{R}^{n})$ implies that there exists $c_{5}=c_{5}(p,n,\alpha _{1},\varphi )>0$ (independent of k) such that

$$ \int _{\mathbb{R}^{n}} \bigl\vert f_{k}\bigl(x,u_{k}(t,x) \bigr) \bigr\vert ^{2}\,dx\leq c_{5} \bigl(1+ \bigl\Vert u_{k}(t) \bigr\Vert ^{2(p+1)}_{H^{2}(\mathbb{R}^{n})} \bigr). $$

(3.35)

By (3.33) and (3.35) we see that

$$ \bigl\{ f_{k}(\cdot ,u_{k})\bigr\} ^{\infty}_{k=1} \quad \text{is bounded in } L^{2}\bigl( \tau ,\tau +T;L^{2}\bigl( \mathbb{R}^{n}\bigr)\bigr). $$

(3.36)

By (3.20) we obtain

$$ \int _{\mathbb{R}^{n}} \bigl\vert h_{k}\bigl(t,x,u_{k}(t,x) \bigr) \bigr\vert ^{2}\,dx\leq 2\alpha _{2} \Vert u_{k} \Vert ^{2}+2 \bigl\Vert \varphi _{2}(t) \bigr\Vert ^{2}, $$

which together with (3.33) shows that

$$ \bigl\{ h_{k}(\cdot ,\cdot ,u_{k})\bigr\} ^{\infty}_{k=1} \quad \text{is bounded in } L^{2}\bigl(\tau , \tau +T;L^{2}\bigl(\mathbb{R}^{n}\bigr)\bigr). $$

(3.37)

By (3.33), (3.34), (3.36), and (3.37), it follows that there exists $u\in L^{\infty}(\tau ,\tau +T;H^{2}(\mathbb{R}^{n}))$ with $\partial _{t}u\in L^{\infty}(\tau ,\tau +T;L^{2}(\mathbb{R}^{n}))$, $\kappa _{1}\in L^{2}(\tau ,\tau +T;L^{2}(\mathbb{R}^{n}))$, $\kappa _{2} \in L^{2}(\tau ,\tau +T;L^{2}(\mathbb{R}^{n}))$, $v^{\tau +T}\in H^{2}( \mathbb{R}^{n})$ and $v^{\tau +T}_{1}\in L^{2}(\mathbb{R}^{n})$ such that

$$\begin{aligned} &u_{k}\rightarrow u\quad \text{weak-star in } L^{\infty}\bigl(\tau , \tau +T;H^{2}\bigl( \mathbb{R}^{n}\bigr)\bigr), \end{aligned}$$

(3.38)

$$\begin{aligned} &\partial _{t}u_{k}\rightarrow \partial _{t}u \quad \text{weak-star in } L^{\infty}\bigl(\tau ,\tau +T;L^{2}\bigl( \mathbb{R}^{n}\bigr)\bigr), \end{aligned}$$

(3.39)

$$\begin{aligned} &f_{k}(\cdot ,u_{k})\rightarrow \kappa _{1} \quad \text{weakly in } L^{2}\bigl( \tau ,\tau +T;L^{2}\bigl( \mathbb{R}^{n}\bigr)\bigr), \end{aligned}$$

(3.40)

$$\begin{aligned} &h_{k}(\cdot ,\cdot ,u_{k})\rightarrow \kappa _{2} \quad \text{weakly in } L^{2}\bigl(\tau ,\tau +T;L^{2}\bigl(\mathbb{R}^{n}\bigr)\bigr), \end{aligned}$$

(3.41)

$$\begin{aligned} &u_{k}(\tau +T)\rightarrow v^{\tau +T} \quad \text{weakly in } H^{2}\bigl( \mathbb{R}^{n}\bigr), \end{aligned}$$

(3.42)

$$\begin{aligned} &\partial _{t}u_{k}(\tau +T)\rightarrow v^{\tau +T}_{1} \quad \text{weakly in } L^{2}\bigl( \mathbb{R}^{n}\bigr). \end{aligned}$$

(3.43)

It follows from (3.38) and (3.39) that there exists a subsequence that is still denoted $u_{k}$, such that

$$ u_{k}(t,x)\rightarrow u(t,x) \quad \text{for almost all } (t,x)\in [ \tau , \tau +T]\times \mathbb{R}^{n}. $$

(3.44)

By (3.13) and (3.44) we obtain that for almost all $(t,x)\in [\tau ,\tau +T]\times \mathbb{R}^{n}$,

$$\begin{aligned} & \bigl\vert \eta _{k}\bigl(u_{k}(t,x)\bigr)-u(t,x) \bigr\vert \\ &\quad \leq \bigl\vert \eta _{k}\bigl(u_{k}(t,x) \bigr)-\eta _{k}\bigl(u(t,x)\bigr) \bigr\vert + \bigl\vert \eta _{k}\bigl(u(t,x)\bigr)-u(t,x) \bigr\vert \\ &\quad \leq \bigl\vert u_{k}(t,x)-u(t,x) \bigr\vert + \bigl\vert \eta _{k}\bigl(u(t,x)\bigr)-u(t,x) \bigr\vert \rightarrow 0,\quad \text{as } k\rightarrow \infty . \end{aligned}$$

(3.45)

By (3.45), we have

$$\begin{aligned} &f_{k}\bigl(x,u_{k}(t,x)\bigr)\rightarrow f\bigl(x,u(t,x) \bigr)\quad \text{for almost all } (t,x)\in [\tau ,\tau +T]\times \mathbb{R}^{n}, \end{aligned}$$

(3.46)

$$\begin{aligned} &h_{k}\bigl(t,x,u_{k}(t,x)\bigr)\rightarrow h \bigl(t,x,u(t,x)\bigr) \quad \text{for almost all } (t,x)\in [\tau ,\tau +T]\times \mathbb{R}^{n}. \end{aligned}$$

(3.47)

It follows from (3.40), (3.41), (3.46), and (3.47) that

$$\begin{aligned} &f_{k}(\cdot ,u_{k})\rightarrow f(\cdot ,u) \quad \text{weakly in } L^{2}\bigl( \tau ,\tau +T;L^{2}\bigl( \mathbb{R}^{n}\bigr)\bigr), \end{aligned}$$

(3.48)

$$\begin{aligned} &h_{k}(\cdot ,\cdot ,u_{k})\rightarrow h(\cdot ,\cdot ,u) \quad \text{weakly in } L^{2}\bigl(\tau ,\tau +T;L^{2}\bigl( \mathbb{R}^{n}\bigr)\bigr). \end{aligned}$$

(3.49)

Step (iii): Existence of solutions

Choosing an arbitrary $\xi \in C^{\infty}_{0}((\tau ,\tau +T)\times \mathbb{R}^{n})$. By (3.24) we obtain

$$\begin{aligned} &- \int ^{\tau +T}_{\tau}(\partial _{t}u_{k}, \xi _{t})\,dt+\alpha \int ^{ \tau +T}_{\tau}(\partial _{t}u_{k}, \xi )\,dt \\ &\qquad {}+ \int ^{\tau +T}_{\tau }( \Delta u_{k},\Delta \xi )\,dt+\nu \int ^{\tau +T}_{\tau}(u_{k},\xi )\,dt \\ &\qquad {} + \int ^{\tau +T}_{\tau} \int _{\mathbb{R}^{n}}f_{k}\bigl(x,u_{k}(t,x)\bigr) \xi (t,x)\,dx\,dt \\ &\quad = \int ^{\tau +T}_{\tau}\bigl(g(t),\xi \bigr)\,dt+ \int ^{\tau +T}_{\tau} \int _{ \mathbb{R}^{n}}h_{k}\bigl(t,x,u_{k}(t,x)\bigr) \zeta _{\delta}(\theta _{t} \omega )\xi (t,x)\,dx\,dt. \end{aligned}$$

(3.50)

Letting $k\rightarrow \infty$ in (3.50), it follows from (3.38), (3.39), (3.48), and (3.49) that for any $\xi \in C^{\infty}_{0}((\tau ,\tau +T)\times \mathbb{R}^{n})$,

$$\begin{aligned} &- \int ^{\tau +T}_{\tau}(u_{t},\xi _{t})\,dt+\alpha \int ^{\tau +T}_{ \tau}(u_{t},\xi )\,dt+ \int ^{\tau +T}_{\tau }(\Delta u,\Delta \xi )\,dt+ \nu \int ^{\tau +T}_{\tau}(u,\xi )\,dt \\ &\qquad {} + \int ^{\tau +T}_{\tau} \int _{\mathbb{R}^{n}}f\bigl(x,u(t,x)\bigr) \xi (t,x)\,dx\,dt \\ &\quad = \int ^{\tau +T}_{\tau}\bigl(g(t),\xi \bigr)\,dt+ \int ^{\tau +T}_{\tau} \int _{ \mathbb{R}^{n}}h\bigl(t,x,u(t,x)\bigr)\zeta _{\delta}(\theta _{t}\omega )\xi (t,x)\,dx\,dt. \end{aligned}$$

(3.51)

Note that

$$ u\in L^{\infty}\bigl(\tau ,\tau +T;H^{2}\bigl( \mathbb{R}^{n}\bigr)\bigr)\quad \text{and}\quad \partial _{t}u\in L^{\infty}\bigl(\tau ,\tau +T;L^{2}\bigl(\mathbb{R}^{n} \bigr)\bigr). $$

(3.52)

By (3.52) we obtain

$$ h(\cdot ,\cdot ,u)\in L^{2}\bigl(\tau ,\tau +T;L^{2}\bigl( \mathbb{R}^{n}\bigr)\bigr). $$

(3.53)

We claim that

$$ f(\cdot ,u)\quad \text{belongs to } L^{\infty}\bigl(\tau ,\tau +T;L^{2} \bigl( \mathbb{R}^{n}\bigr)\bigr). $$

(3.54)

In fact, by (3.3) we obtain that there exists some $c_{6}=c_{6}(p,n,\alpha _{1},\varphi )>0$ such that

$$\begin{aligned} \bigl\Vert f\bigl(\cdot ,u(t)\bigr) \bigr\Vert ^{2}&\leq 2\alpha ^{2}_{1} \bigl( \Vert \varphi \Vert ^{2}_{L^{ \infty}(\mathbb{R}^{n})} \bigl\Vert u(t) \bigr\Vert ^{2} + \bigl\Vert u(t) \bigr\Vert ^{2(p+1)}_{L^{2(p+1)}( \mathbb{R}^{n})} \bigr) \\ &\leq c_{6} \bigl( \bigl\Vert u(t) \bigr\Vert ^{2}+ \bigl\Vert u(t) \bigr\Vert ^{2(p+1)}_{H^{2}(\mathbb{R}^{n})} \bigr), \end{aligned}$$

which along with (3.52) leads to (3.54).

By (3.51)–(3.54), we can obtain

$$ u_{tt}\quad \text{belongs to } L^{2}\bigl(\tau ,\tau +T;H^{-2}\bigl( \mathbb{R}^{n}\bigr)\bigr), $$

(3.55)

where $H^{-2}(\mathbb{R}^{n})$ is the dual space of $H^{2}(\mathbb{R}^{n})$.

Next, we prove u and $u_{t}$ satisfy the initial conditions (1.1)₂.

By (3.24), we obtain that for any $v\in C^{\infty}_{0}( \mathbb{R}^{n})$ and $\psi \in C^{2}([\tau ,\tau +T])$,

$$\begin{aligned} & \int ^{\tau +T}_{\tau}\bigl(u_{k}(t),v\bigr)\psi ''(t)\,dt+\bigl(\partial _{t}u_{k}( \tau +T),v\bigr)\psi (\tau +T) -\bigl(u_{k}(\tau +T),v\bigr)\psi '(\tau +T) \\ &\qquad {}+(u_{0},v)\psi '(\tau )-(u_{1,0},v)\psi (\tau ) +\alpha \int ^{\tau +T}_{ \tau}\bigl(\partial _{t}u_{k}(t),v \bigr)\psi (t)\,dt \\ &\qquad {}+ \int ^{\tau +T}_{\tau }\bigl( \Delta u_{k}(t), \Delta v\bigr)\psi (t)\,dt \\ &\qquad {}+\nu \int ^{\tau +T}_{\tau}\bigl(u_{k}(t),v\bigr)\psi (t)\,dt+ \int ^{\tau +T}_{ \tau} \int _{\mathbb{R}^{n}}f_{k}\bigl(x,u_{k}(t,x) \bigr)v(x)\psi (t)\,dx\,dt \\ &\quad = \int ^{\tau +T}_{\tau}\bigl(g(t),v\bigr)\psi (t)\,dt+ \int ^{\tau +T}_{\tau} \int _{\mathbb{R}^{n}}h_{k}\bigl(t,x,u_{k}(t,x)\bigr) \zeta _{\delta}(\theta _{t} \omega )v(x)\psi (t)\,dx\,dt. \end{aligned}$$

(3.56)

Letting $k\rightarrow \infty$ in (3.56), by (3.38), (3.39), (3.42), (3.43), (3.48), and (3.49) we obtain, for any $v\in C^{\infty}_{0}(\mathbb{R}^{n})$ and $\psi \in C^{2}([\tau ,\tau +T])$,

$$\begin{aligned} & \int ^{\tau +T}_{\tau}\bigl(u(t),v\bigr)\psi ''(t)\,dt+\bigl(v^{\tau +T}_{1},v\bigr) \psi ( \tau +T) -\bigl(v^{\tau +T},v\bigr)\psi '(\tau +T) \\ &\qquad {}+(u_{0},v)\psi '(\tau )-(u_{1,0},v)\psi (\tau ) +\alpha \int ^{\tau +T}_{ \tau}\bigl(\partial _{t}u(t),v \bigr)\psi (t)\,dt \\ &\qquad {}+ \int ^{\tau +T}_{\tau }\bigl(\Delta u(t), \Delta v\bigr)\psi (t)\,dt \\ &\qquad {}+\nu \int ^{\tau +T}_{\tau}\bigl(u(t),v\bigr)\psi (t)\,dt+ \int ^{\tau +T}_{\tau} \int _{\mathbb{R}^{n}}f\bigl(x,u(t,x)\bigr)v(x)\psi (t)\,dx\,dt \\ &\quad = \int ^{\tau +T}_{\tau}\bigl(g(t),v\bigr)\psi (t)\,dt+ \int ^{\tau +T}_{\tau} \int _{\mathbb{R}^{n}}h\bigl(t,x,u(t,x)\bigr)\zeta _{\delta}(\theta _{t}\omega )v(x) \psi (t)\,dx\,dt. \end{aligned}$$

(3.57)

By (3.51) we obtain that for any $v\in C^{\infty}_{0}(\mathbb{R}^{n})$,

$$\begin{aligned} &\frac{d}{dt}(u_{t},v)+\alpha (u_{t},v) + (\Delta u, \Delta v) +\nu (u,v) + \int _{\mathbb{R}^{n}}f\bigl(x,u(t,x)\bigr)v(x)\,dx \\ &\quad = \bigl(g(t),v\bigr) + \int _{\mathbb{R}^{n}}h\bigl(t,x,u(t,x)\bigr)\zeta _{\delta}( \theta _{t}\omega )v(x)\,dx. \end{aligned}$$

(3.58)

By (3.58) we find that for any $v\in C^{\infty}_{0}(\mathbb{R}^{n})$ and $\psi \in C^{2}([\tau ,\tau +T])$,

$$\begin{aligned} & \int ^{\tau +T}_{\tau}\bigl(u(t),v\bigr)\psi ''(t)\,dt+\bigl(\partial _{t}u(\tau +T),v \bigr) \psi (\tau +T) -\bigl(u(\tau +T),v\bigr)\psi '(\tau +T) \\ &\qquad {}+\bigl(u(\tau ),v\bigr)\psi '(\tau )-\bigl(\partial _{t}u(\tau ),v\bigr)\psi (\tau ) + \alpha \int ^{\tau +T}_{\tau}\bigl(\partial _{t}u(t),v \bigr)\psi (t)\,dt \\ &\qquad {}+ \int ^{ \tau +T}_{\tau }\bigl(\Delta u(t),\Delta v\bigr)\psi (t)\,dt \\ &\qquad {}+\nu \int ^{\tau +T}_{\tau}\bigl(u(t),v\bigr)\psi (t)\,dt+ \int ^{\tau +T}_{\tau} \int _{\mathbb{R}^{n}}f\bigl(x,u(t,x)\bigr)v(x)\psi (t)\,dx\,dt \\ &\quad = \int ^{\tau +T}_{\tau}\bigl(g(t,\cdot ),v\bigr)\psi (t)\,dt+ \int ^{\tau +T}_{ \tau} \int _{\mathbb{R}^{n}}h\bigl(t,x,u(t,x)\bigr)\zeta _{\delta}(\theta _{t} \omega )v(x)\psi (t)\,dx\,dt, \end{aligned}$$

(3.59)

together with (3.57) to obtain, for $v\in C^{\infty}_{0}(\mathbb{R}^{n})$ and $\psi \in C^{2}([\tau ,\tau +T])$,

$$\begin{aligned} &\bigl(v^{\tau +T}_{1},v\bigr)\psi (\tau +T) - \bigl(v^{\tau +T},v\bigr)\psi '(\tau +T) +(u_{0},v) \psi '(\tau )-(u_{1,0},v)\psi (\tau ) \\ &\quad =\bigl(\partial _{t}u(\tau +T),v\bigr)\psi (\tau +T) -\bigl(u(\tau +T),v\bigr)\psi '( \tau +T) +\bigl(u(\tau ),v\bigr)\psi '(\tau ) \\ &\qquad {}-\bigl(\partial _{t}u(\tau ),v\bigr)\psi ( \tau ). \end{aligned}$$

(3.60)

Let $\psi \in C^{2}([\tau ,\tau +T])$ such that $\psi (\tau +T)=\psi '(\tau +T)=\psi '(\tau )=0$ and $\psi (\tau )=1$, then by (3.60) we have

$$ \bigl(\partial _{t}u(\tau ),v\bigr)=(u_{1,0},v), \quad \forall v \in C^{\infty}_{0}\bigl( \mathbb{R}^{n}\bigr). $$

(3.61)

Let $\psi \in C^{2}([\tau ,\tau +T])$ such that $\psi (\tau +T)=\psi '(\tau +T)=\psi (\tau )=0$ and $\psi '(\tau )=1$, then by (3.60) we have

$$ \bigl(u(\tau ),v\bigr)=(u_{0},v), \quad \forall v\in C^{\infty}_{0} \bigl(\mathbb{R}^{n}\bigr), $$

(3.62)

which together with (3.61) shows that u satisfies the initial conditions (1.1)₂.

Through choosing proper $\psi \in C^{2}([\tau ,\tau +T])$, we can also obtain from (3.60) that

$$ u(\tau +T)=v^{\tau +T}, \quad \text{and}\quad \partial _{t}u(\tau +T)=v^{ \tau +T}_{1}, $$

which along with (3.42) and (3.43) implies that

$$\begin{aligned} &u_{k}(\tau +T)\rightarrow u(\tau +T)\quad \text{weakly in } H^{2}\bigl( \mathbb{R}^{n}\bigr), \end{aligned}$$

(3.63)

$$\begin{aligned} &\partial _{t}u_{k}(\tau +T)\rightarrow \partial _{t}u(\tau +T) \quad \text{weakly in } L^{2}\bigl( \mathbb{R}^{n}\bigr). \end{aligned}$$

(3.64)

Similar to (3.63) and (3.64), one can verify that for any $t\in [\tau , \tau +T]$,

$$\begin{aligned} &u_{k}(t)\rightarrow u(t) \quad \text{weakly in } H^{2}\bigl( \mathbb{R}^{n}\bigr), \end{aligned}$$

(3.65)

$$\begin{aligned} &\partial _{t}u_{k}(t)\rightarrow \partial _{t}u(t)\quad \text{weakly in } L^{2}\bigl(\mathbb{R}^{n} \bigr). \end{aligned}$$

(3.66)

Thus, we obtain the claim. By (3.65) and (3.66), we obtain that u is a solution of (1.1) in the sense of Definition 3.1.

Step (iv): Uniqueness of solutions

Let $u_{1}$ and $u_{2}$ be solutions to (1.1), denote $v=u_{1}-u_{2}$. Then, we have

$$ \textstyle\begin{cases} v_{tt}+\alpha v_{t}+\Delta ^{2}v+\nu v =f(\cdot ,u_{2})-f(\cdot ,u_{1})+(h(t, \cdot ,u_{1})-h(t,\cdot ,u_{2}))\zeta _{\delta}(\theta _{t}\omega ), \\ v(\tau )=0, \qquad v_{t}(\tau )=0. \end{cases} $$

(3.67)

By (3.8), we obtain

$$\begin{aligned} &\frac{d}{dt}\bigl( \Vert v_{t} \Vert ^{2}+ \Vert \Delta v \Vert ^{2}+\nu \Vert v \Vert ^{2}\bigr) \\ &\quad =-2\alpha \Vert v_{t} \Vert ^{2} +2\bigl(f(\cdot ,u_{2})-f(\cdot ,u_{1}),v_{t}\bigr)+2\bigl(h(t, \cdot ,u_{1})-h(t,\cdot ,u_{2}),v_{t}\bigr)\zeta _{\delta}(\theta _{t} \omega ). \end{aligned}$$

(3.68)

Since $H^{2}(\mathbb{R}^{n})\hookrightarrow L^{2(p+1)}(\mathbb{R}^{n})$ for $0< p\leq \frac{4}{n-4}$, by (3.3), we obtain

$$ \bigl\Vert f(\cdot ,u_{2})-f(\cdot ,u_{1}) \bigr\Vert \leq \alpha _{1} \Vert \varphi \Vert _{L^{ \infty}(\mathbb{R}^{n})} \Vert v \Vert +\alpha _{1} \bigl( \Vert u_{1} \Vert ^{p}_{H^{2}( \mathbb{R}^{n})}+ \Vert u_{2} \Vert ^{p}_{H^{2}(\mathbb{R}^{n})} \bigr) \Vert v \Vert _{H^{2}( \mathbb{R}^{n})} $$

and hence

$$\begin{aligned} &2\bigl(f(\cdot ,u_{2})-f(\cdot ,u_{1}),v_{t} \bigr) \\ &\quad \leq 2 \bigl\Vert f(\cdot ,u_{2})-f(\cdot ,u_{1}) \bigr\Vert \Vert v_{t} \Vert \\ &\quad \leq\alpha _{1} \bigl( \Vert \varphi \Vert _{L^{\infty}(\mathbb{R}^{n})}+ \Vert u_{1} \Vert ^{p}_{H^{2}(\mathbb{R}^{n})}+ \Vert u_{2} \Vert ^{p}_{H^{2}(\mathbb{R}^{n})} \bigr) \bigl( \Vert v \Vert ^{2}_{H^{2}(\mathbb{R}^{n})}+ \Vert v_{t} \Vert ^{2} \bigr). \end{aligned}$$

(3.69)

By (3.6) we obtain

$$\begin{aligned} &2\bigl(h(t,\cdot ,u_{1})-h(t,\cdot ,u_{2}),v_{t} \bigr)\zeta _{\delta}(\theta _{t} \omega ) \\ &\quad \leq \bigl\Vert h(t,\cdot ,u_{1})-h(t,\cdot ,u_{2}) \bigr\Vert \Vert v_{t} \Vert \bigl\vert \zeta _{\delta}( \theta _{t}\omega ) \bigr\vert \\ &\quad \leq2\alpha _{3} \Vert v \Vert \Vert v_{t} \Vert \bigl\vert \zeta _{\delta}(\theta _{t}\omega ) \bigr\vert \\ &\quad \leq\alpha _{3}\bigl( \Vert v \Vert ^{2}+ \Vert v_{t} \Vert ^{2}\bigr) \bigl\vert \zeta _{\delta}( \theta _{t} \omega ) \bigr\vert . \end{aligned}$$

(3.70)

It follows from (3.68)–(3.70) that

$$\begin{aligned} &\frac{d}{dt}\bigl( \Vert v_{t} \Vert ^{2}+ \Vert \Delta v \Vert ^{2}+\nu \Vert v \Vert ^{2}\bigr) \\ &\quad \leq c_{7} \bigl(1+ \Vert u_{1} \Vert ^{p}_{H^{2}(\mathbb{R}^{n})}+ \Vert u_{2} \Vert ^{p}_{H^{2}( \mathbb{R}^{n})} \bigr) \bigl( \Vert v_{t} \Vert ^{2}+ \Vert \Delta v \Vert ^{2}+\nu \Vert v \Vert ^{2} \bigr), \end{aligned}$$

(3.71)

where $c_{7} > 0$ depends on τ and T. Since $u_{1}, u_{2}\in L^{\infty}(\tau ,\tau +T;H^{2}(\mathbb{R}^{n}))$, then applying Gronwall’s lemma on $[\tau ,\tau +T]$, we can obtain the uniqueness of solution as well as the continuous dependence property of the solution with initial data. □

We now define a mapping $\Phi :\mathbb{R}^{+}\times \mathbb{R}\times \Omega \times H^{2}( \mathbb{R}^{n})\times L^{2}(\mathbb{R}^{n})\rightarrow H^{2}( \mathbb{R}^{n})\times L^{2}(\mathbb{R}^{n})$ such that for all $t\in \mathbb{R}^{+}$, $\tau \in \mathbb{R}$, $\omega \in \Omega$ and $(u_{0},u_{1,0})\in H^{2}(\mathbb{R}^{n})\times L^{2}(\mathbb{R}^{n})$,

$$ \Phi \bigl(t,\tau ,\omega ,(u_{0},u_{1,0})\bigr)=\bigl(u(t+ \tau ,\tau ,\theta _{- \tau}\omega ,u_{0}),u_{t}(t+ \tau ,\tau ,\theta _{-\tau}\omega ,u_{1,0})\bigr), $$

(3.72)

where u is the solution of (1.1). Then, Φ is a continuous cocycle on $H^{2}(\mathbb{R}^{n})\times L^{2}(\mathbb{R}^{n})$ over $(\Omega ,\mathcal{F},P,\{\theta _{t}\}_{t\in \mathbb{R}})$.

4 Uniform estimates of solutions

In this section, we derive necessary estimates of solutions of (1.1) under stronger conditions than (3.2)–(3.6) on the nonlinear functions f and h. These estimates are useful for proving the asymptotic compactness of the solutions and the existence of pullback random attractors.

From now on, we assume f satisfies: for all $x\in \mathbb{R}^{n}$ and $s\in \mathbb{R}$,

$$\begin{aligned} &f(x,s)s-\gamma F(x,s)\geq \varphi _{3}(x), \end{aligned}$$

(4.1)

$$\begin{aligned} &F(x,s)+\varphi _{1}(x)\geq \alpha _{4} \vert s \vert ^{p+2}, \end{aligned}$$

(4.2)

$$\begin{aligned} & \bigl\vert \partial _{s}f(x,s) \bigr\vert \leq \iota \vert s \vert ^{p}+\varsigma , \qquad \bigl\vert \partial _{x}f(x,s) \bigr\vert \leq \varphi _{4}(x), \end{aligned}$$

(4.3)

where $p>0$ for $1\leq n\leq 4$ and $0< p\leq \frac{4}{n-4}$ for $n\geq 5$, $\gamma \in (0,1]$, $\alpha _{4}$, ς are positive constants, $\varphi _{3}\in L^{1}(\mathbb{R}^{n})$, and $\varphi _{4}\in L^{2}(\mathbb{R}^{n})\cap L^{\infty}(\mathbb{R}^{n})$, $\iota >0$ will be denoted later.

By (3.3) and (4.1) we obtain that for all $x\in \mathbb{R}^{n}$ and $s\in \mathbb{R}$,

$$ \gamma F(x,s)\leq \alpha _{1}s^{2}\varphi (x)+\alpha _{1} \vert s \vert ^{p+2}- \varphi _{3}(x). $$

(4.4)

Assume the nonlinearity h satisfies: for all $x\in \mathbb{R}^{n}$ and $t, s\in \mathbb{R}$,

$$\begin{aligned} & \bigl\vert h(t,x,s) \bigr\vert \leq \varphi _{5}(x) \vert s \vert +\varphi _{6}(x), \end{aligned}$$

(4.5)

$$\begin{aligned} & \bigl\vert \partial _{x}h(t,x,s) \bigr\vert + \bigl\vert \partial _{s}h(t,x,s) \bigr\vert \leq \varphi _{7}(x), \end{aligned}$$

(4.6)

where $\varphi _{5}\in L^{\infty}(\mathbb{R}^{n})\cap L^{2+\frac{4}{p}}( \mathbb{R}^{n})$, $\varphi _{6}\in L^{2}(\mathbb{R}^{n})$, and $\varphi _{7}\in L^{2}(\mathbb{R}^{n})\cap L^{\infty}(\mathbb{R}^{n})$.

Let $\mathcal{D}$ be the set of all tempered families of nonempty bounded subsets of $H^{2}(\mathbb{R}^{n})\times L^{2}(\mathbb{R}^{n})$. $D=\{D(\tau ,\Omega ):\tau \in \mathbb{R},\omega \in \Omega \}$ is called tempered if for any $c>0$,

$$ \lim_{t\rightarrow +\infty}e^{-ct} \bigl\Vert D(\tau -t,\theta _{-t}\omega ) \bigr\Vert _{H^{2}( \mathbb{R}^{n})\times L^{2}(\mathbb{R}^{n})}=0, $$

where $\|D\|_{H^{2}(\mathbb{R}^{n})\times L^{2}(\mathbb{R}^{n})}=\sup_{\xi \in D}\|\xi \|_{H^{2}(\mathbb{R}^{n})\times L^{2}( \mathbb{R}^{n})}$.

Under $\alpha >0$, $\nu >0$, and $\gamma \in (0, 1]$, we can choose a sufficiently small positive constant ε such that

$$ \begin{aligned}&\varepsilon < \min \biggl\{ 1,\nu ,\frac{2\alpha}{5}\biggr\} ,\qquad \frac{1}{2} \alpha -2 \varepsilon -\frac{1}{8}\varepsilon \gamma >0, \\ & \nu - \frac{1}{2}\nu \gamma -\varepsilon \alpha +\frac{1}{8}\varepsilon ^{2}\gamma >0, \qquad \nu -\varepsilon -\varepsilon \alpha +\frac{1}{2} \varepsilon ^{2}>0. \end{aligned}$$

(4.7)

We also assume

$$\begin{aligned} & \int ^{\tau}_{-\infty}e^{\frac{1}{4}\varepsilon \gamma s} \bigl\Vert g(s) \bigr\Vert ^{2}_{1}\,ds< \infty , \quad \forall \tau \in \mathbb{R}, \end{aligned}$$

(4.8)

$$\begin{aligned} &\lim_{t\rightarrow +\infty}e^{-ct} \int ^{0}_{-\infty}e^{\frac{1}{4} \varepsilon \gamma s} \bigl\Vert g(s-t) \bigr\Vert ^{2}_{1}\,ds=0, \quad\text{for } \forall c>0. \end{aligned}$$

(4.9)

Lemma 4.1

Let (3.2), (3.3), (3.6), (4.1), (4.2), and (4.5)–(4.8) hold. Then, for any $\tau \in \mathbb{R}$, $\omega \in \Omega$ and $D\in \mathcal{D}$, there exists $T=T(\tau ,\omega ,D)>0$ such that for all $t\geq T$, the solution of (1.1) satisfies

$$\begin{aligned} & \bigl\Vert u_{t}(\tau ,\tau -t,\theta _{-\tau}\omega ,u_{1,0}) \bigr\Vert ^{2}+ \bigl\Vert u( \tau ,\tau -t, \theta _{-\tau}\omega ,u_{0}) \bigr\Vert ^{2}_{H^{2}(\mathbb{R}^{n})} \\ &\qquad {}+ \int ^{\tau}_{\tau -t}e^{\frac{1}{4}\varepsilon \gamma (s-\tau )}\bigl( \bigl\Vert u_{t}(s,\tau -t,\theta _{-\tau}\omega ,u_{1,0}) \bigr\Vert ^{2}+ \bigl\Vert u(s,\tau -t, \theta _{-\tau}\omega ,u_{0}) \bigr\Vert ^{2}_{H^{2}(\mathbb{R}^{n})}\bigr)\,ds \\ &\quad \leq M_{1}+M_{1} \int ^{0}_{-\infty}e^{\frac{1}{4}\varepsilon \gamma s}\bigl(1+ \bigl\Vert g(s+\tau ) \bigr\Vert ^{2}+ \bigl\vert \zeta _{\delta}(\theta _{s}\omega ) \bigr\vert ^{2+ \frac{4}{p}}\bigr)\,ds, \end{aligned}$$

where $(u_{0},u_{1,0}) \in D(\tau -t,\theta _{-t}\omega )$ and $M_{1}$ is a positive constant independent of τ, ω, and D.

Proof

By (3.9), (3.11), (4.1), and (4.10) we obtain, for almost all $t\in [\tau ,\tau +T]$,

$$\begin{aligned} &\frac{d}{dt}\biggl( \Vert u_{t} \Vert ^{2}+\nu \Vert u \Vert ^{2}+ \Vert \Delta u \Vert ^{2}+2 \int _{ \mathbb{R}^{n}}F\bigl(x,u(t,x)\bigr)\,dx+\varepsilon (u,u_{t})\biggr) \\ &\qquad {}+(2\alpha -\varepsilon ) \Vert u_{t} \Vert ^{2}+ \varepsilon \alpha (u,u_{t})+ \varepsilon \Vert \Delta u \Vert ^{2}+\varepsilon \nu \Vert u \Vert ^{2}+\varepsilon \gamma \int _{\mathbb{R}}F\bigl(x,u(t,x)\bigr)\,dx \\ &\quad \leq\varepsilon \Vert \varphi _{3} \Vert _{L^{1}(\mathbb{R}^{n})}+ \bigl(g(t)+h\bigl(t, \cdot ,u(t)\bigr)\zeta _{\delta}(\theta _{t} \omega ),\varepsilon u+2u_{t}\bigr). \end{aligned}$$

(4.10)

For the second term on the right-hand side of (4.10), using (4.2) and (4.5) we have

$$\begin{aligned} &\bigl(g(t)+h\bigl(t,\cdot ,u(t)\bigr)\zeta _{\delta}(\theta _{t}\omega ), \varepsilon u+2u_{t}\bigr) \\ &\quad \leq\bigl( \bigl\Vert g(t) \bigr\Vert + \bigl\Vert h\bigl(t,\cdot ,u(t) \bigr)\zeta _{\delta}(\theta _{t}\omega ) \bigr\Vert \bigr) \bigl(\varepsilon \Vert u \Vert +2 \Vert u_{t} \Vert \bigr) \\ &\quad \leq\frac{1}{2}\varepsilon \nu \Vert u \Vert ^{2}+\alpha \Vert u_{t} \Vert ^{2}+\biggl( \alpha ^{-1}+ \frac{1}{2}\varepsilon \nu ^{-1}\biggr) \bigl( \bigl\Vert g(t) \bigr\Vert + \bigl\Vert h\bigl(t,\cdot ,u(t)\bigr) \zeta _{\delta}( \theta _{t}\omega ) \bigr\Vert \bigr)^{2} \\ &\quad \leq\frac{1}{2}\varepsilon \nu \Vert u \Vert ^{2}+\alpha \Vert u_{t} \Vert ^{2}+\bigl(2 \alpha ^{-1} + \varepsilon \nu ^{-1}\bigr) \bigl\Vert g(t) \bigr\Vert ^{2}+\bigl(2\alpha ^{-1}+ \varepsilon \nu ^{-1} \bigr) \bigl\Vert h\bigl(t,\cdot ,u(t)\bigr)\zeta _{\delta}(\theta _{t} \omega ) \bigr\Vert ^{2} \\ &\quad \leq\frac{1}{2}\varepsilon \nu \Vert u \Vert ^{2}+\alpha \Vert u_{t} \Vert ^{2}+\bigl(2 \alpha ^{-1} + \varepsilon \nu ^{-1}\bigr) \bigl\Vert g(t) \bigr\Vert ^{2}+2\bigl(2\alpha ^{-1}+ \varepsilon \nu ^{-1} \bigr) \bigl\vert \zeta _{\delta}(\theta _{t}\omega ) \bigr\vert ^{2} \Vert \varphi _{6} \Vert ^{2} \\ &\qquad {}+2\bigl(2\alpha ^{-1}+\varepsilon \nu ^{-1}\bigr) \bigl\vert \zeta _{\delta}(\theta _{t} \omega ) \bigr\vert ^{2} \int _{\mathbb{R}^{n}} \bigl\vert \varphi _{5}(x) \bigr\vert ^{2} \bigl\vert u(t,x) \bigr\vert ^{2}\,dx \\ &\quad \leq\frac{1}{2}\varepsilon \nu \Vert u \Vert ^{2}+\alpha \Vert u_{t} \Vert ^{2}+\bigl(2 \alpha ^{-1} + \varepsilon \nu ^{-1}\bigr) \bigl\Vert g(t) \bigr\Vert ^{2}+c_{1} \bigl\vert \zeta _{\delta}( \theta _{t}\omega ) \bigr\vert ^{2} \\ &\qquad {}+\frac{1}{2}\varepsilon \gamma \alpha _{4} \int _{\mathbb{R}^{n}} \bigl\vert u(t,x) \bigr\vert ^{p+2}\,dx \\ &\qquad {}+c_{2} \bigl\vert \zeta _{\delta}(\theta _{t} \omega ) \bigr\vert ^{2+\frac{4}{p}} \int _{ \mathbb{R}^{n}} \bigl\vert \varphi _{5}(x) \bigr\vert ^{2+\frac{4}{p}}\,dx \\ &\quad \leq\frac{1}{2}\varepsilon \nu \Vert u \Vert ^{2}+\alpha \Vert u_{t} \Vert ^{2}+\bigl(2 \alpha ^{-1} + \varepsilon \nu ^{-1}\bigr) \bigl\Vert g(t) \bigr\Vert ^{2}+c_{1} \bigl\vert \zeta _{\delta}( \theta _{t}\omega ) \bigr\vert ^{2} \\ &\qquad {}+\frac{1}{2}\varepsilon \gamma \int _{ \mathbb{R}}F\bigl(x,u(t,x)\bigr)\,dx \\ &\qquad {}+\frac{1}{2}\varepsilon \gamma \Vert \varphi _{1} \Vert _{L^{1}(\mathbb{R}^{n})}+c_{3} \bigl\vert \zeta _{\delta}(\theta _{t}\omega ) \bigr\vert ^{2+\frac{4}{p}} \\ &\quad \leq\frac{1}{2}\varepsilon \nu \Vert u \Vert ^{2}+\alpha \Vert u_{t} \Vert ^{2}+\bigl(2 \alpha ^{-1} + \varepsilon \nu ^{-1}\bigr) \bigl\Vert g(t) \bigr\Vert ^{2} \\ &\qquad {}+\frac{1}{2} \varepsilon \gamma \int _{\mathbb{R}}F\bigl(x,u(t,x)\bigr)\,dx+c_{4}\bigl(1+ \bigl\vert \zeta _{ \delta}(\theta _{t}\omega ) \bigr\vert ^{2+\frac{4}{p}}\bigr), \end{aligned}$$

(4.11)

where $c_{4}>0$ depends on α, ν, γ, ε.

It follows from (4.10) and (4.11) and rewriting the result obtained, we have

$$\begin{aligned} &\frac{d}{dt}\biggl( \Vert u_{t} \Vert ^{2}+\nu \Vert u \Vert ^{2}+ \Vert \Delta u \Vert ^{2}+2 \int _{ \mathbb{R}^{n}}F\bigl(x,u(t,x)\bigr)\,dx+\varepsilon (u,u_{t})\biggr) \\ &\qquad {}+\frac{1}{4}\varepsilon \gamma \biggl( \Vert u_{t} \Vert ^{2}+\nu \Vert u \Vert ^{2}+ \Vert \Delta u \Vert ^{2}+2 \int _{\mathbb{R}^{n}}F\bigl(x,u(t,x)\bigr)\,dx+\varepsilon (u,u_{t})\biggr) \\ &\qquad {}+\biggl(\alpha -\varepsilon -\frac{1}{4}\varepsilon \gamma \biggr) \Vert u_{t} \Vert ^{2}+ \varepsilon \biggl(1- \frac{1}{4}\gamma \biggr) \Vert \Delta u \Vert ^{2}+ \frac{1}{2} \varepsilon \nu \biggl(1-\frac{1}{2}\gamma \biggr) \Vert u \Vert ^{2} \\ &\quad \leq c_{5}\bigl(1+ \bigl\Vert g(t) \bigr\Vert ^{2}+ \bigl\vert \zeta _{\delta}(\theta _{t}\omega ) \bigr\vert ^{2+ \frac{4}{p}}\bigr)-\varepsilon \biggl(\alpha -\frac{1}{4}\varepsilon \gamma \biggr) (u,u_{t})), \end{aligned}$$

(4.12)

where $c_{5}>0$ depends on α, ν, γ, ε.

For the second term on the right-hand side of (4.12) we obtain

$$\begin{aligned} &-\varepsilon \biggl(\alpha -\frac{1}{4}\varepsilon \gamma \biggr) (u,u_{t})) \\ &\quad \leq\varepsilon \biggl(\alpha -\frac{1}{4}\varepsilon \gamma \biggr) \Vert u \Vert \Vert u_{t} \Vert \\ &\quad \leq\frac{1}{2}\varepsilon ^{2}\biggl(\alpha - \frac{1}{4}\varepsilon \gamma \biggr) \Vert u \Vert ^{2}+ \frac{1}{2}\biggl(\alpha -\frac{1}{4}\varepsilon \gamma \biggr) \Vert u_{t} \Vert ^{2}. \end{aligned}$$

(4.13)

By (4.12) and (4.13) we obtain

$$\begin{aligned} &\frac{d}{dt}\biggl( \Vert u_{t} \Vert ^{2}+\nu \Vert u \Vert ^{2}+ \Vert \Delta u \Vert ^{2}+2 \int _{ \mathbb{R}^{n}}F\bigl(x,u(t,x)\bigr)\,dx+\varepsilon (u,u_{t})\biggr) \\ &\qquad {}+\frac{1}{4}\varepsilon \gamma \biggl( \Vert u_{t} \Vert ^{2}+\nu \Vert u \Vert ^{2}+ \Vert \Delta u \Vert ^{2}+2 \int _{\mathbb{R}^{n}}F\bigl(x,u(t,x)\bigr)\,dx+\varepsilon (u,u_{t})\biggr) \\ &\qquad {}+\biggl(\frac{1}{2}\alpha -\varepsilon -\frac{1}{8}\varepsilon \gamma \biggr) \Vert u_{t} \Vert ^{2}+\varepsilon \biggl(1-\frac{1}{4}\gamma \biggr) \Vert \Delta u \Vert ^{2} \\ &\qquad {}+ \frac{1}{2} \varepsilon \biggl(\nu -\frac{1}{2}\nu \gamma - \varepsilon \alpha + \frac{1}{4}\varepsilon ^{2}\gamma \biggr) \Vert u \Vert ^{2} \\ &\quad \leq c_{5}\bigl(1+ \bigl\Vert g(t) \bigr\Vert ^{2}+ \bigl\vert \zeta _{\delta}(\theta _{t}\omega ) \bigr\vert ^{2+ \frac{4}{p}}\bigr). \end{aligned}$$

(4.14)

Multiplying (4.14) by $e^{\frac{1}{4}\varepsilon \gamma t}$, and then integrating the inequality $[\tau -t,\tau ]$, after replacing ω by $\theta _{-\tau}\omega$, we obtain

$$\begin{aligned} & \bigl\Vert u_{t}(\tau ,\tau -t,\theta _{-\tau}\omega ,u_{1,0}) \bigr\Vert ^{2}+\nu \bigl\Vert u( \tau ,\tau -t, \theta _{-\tau}\omega ,u_{0}) \bigr\Vert ^{2}+ \bigl\Vert \Delta u(\tau , \tau -t,\theta _{-\tau}\omega ,u_{0}) \bigr\Vert ^{2} \\ &\qquad {}+2 \int _{\mathbb{R}^{n}}F\bigl(x,u(\tau ,\tau -t,\theta _{-\tau} \omega ,u_{0})\bigr)\,dx \\ &\qquad {} +\varepsilon \bigl(u(\tau ,\tau -t,\theta _{-\tau}\omega ,u_{0}),u_{t}( \tau ,\tau -t,\theta _{-\tau}\omega ,u_{1,0})\bigr) \\ &\qquad {}+\biggl(\frac{1}{2}\alpha -\varepsilon -\frac{1}{8}\varepsilon \gamma \biggr) \int ^{\tau}_{\tau -t}e^{\frac{1}{4}\varepsilon \gamma (s-\tau )} \bigl\Vert u_{t}(s, \tau -t,\theta _{-\tau}\omega ,u_{1,0}) \bigr\Vert ^{2}\,ds \\ &\qquad {}+\varepsilon \biggl(1-\frac{1}{4}\gamma \biggr) \int ^{\tau}_{\tau -t}e^{ \frac{1}{4}\varepsilon \gamma (s-\tau )} \bigl\Vert \Delta u(s,\tau -t,\theta _{- \tau}\omega ,u_{0}) \bigr\Vert ^{2}\,ds \\ &\qquad {}+\frac{1}{2}\varepsilon \biggl(\nu -\frac{1}{2}\nu \gamma - \varepsilon \alpha +\frac{1}{4}\varepsilon ^{2}\gamma \biggr) \int ^{\tau}_{\tau -t}e^{ \frac{1}{4}\varepsilon \gamma (s-\tau )} \bigl\Vert u(s, \tau -t,\theta _{-\tau} \omega ,u_{0}) \bigr\Vert ^{2}\,ds \\ &\quad \leq e^{-\frac{1}{4}\varepsilon \gamma t} \biggl( \Vert u_{1,0} \Vert ^{2}+ \nu \Vert u_{0} \Vert ^{2}+ \Vert \Delta u_{0} \Vert ^{2}+2 \int _{\mathbb{R}^{n}}F(x,u_{0})\,dx+ \varepsilon (u_{0},u_{1,0}) \biggr) \\ &\qquad {}+c_{5} \int ^{\tau}_{\tau -t}e^{\frac{1}{4}\varepsilon \gamma (s- \tau )} \bigl(1+ \bigl\Vert g(s) \bigr\Vert ^{2}+ \bigl\vert \zeta _{\delta}( \theta _{s-\tau}\omega ) \bigr\vert ^{2+ \frac{4}{p}} \bigr)\,ds. \end{aligned}$$

(4.15)

For the first term on the right-hand side of (4.15), by (4.4) we obtain

$$\begin{aligned} &e^{-\frac{1}{4}\varepsilon \gamma t} \biggl( \Vert u_{1,0} \Vert ^{2}+\nu \Vert u_{0} \Vert ^{2}+ \Vert \Delta u_{0} \Vert ^{2}+2 \int _{\mathbb{R}^{n}}F(x,u_{0})\,dx+ \varepsilon (u_{0},u_{1,0}) \biggr) \\ &\quad \leq c_{6}e^{-\frac{1}{4}\varepsilon \gamma t} \bigl(1+ \Vert u_{1,0} \Vert ^{2}+ \Vert u_{0} \Vert ^{2}_{H^{2}(\mathbb{R}^{n}}+ \Vert u_{0} \Vert ^{p+2}_{H^{2}( \mathbb{R}^{n})} \bigr) \\ &\quad \leq c_{7}e^{-\frac{1}{4}\varepsilon \gamma t} \bigl(1+ \bigl\Vert D(\tau -t, \theta _{-t}\omega ) \bigr\Vert ^{p+2}\bigr)\rightarrow 0, \quad \text{as } t \rightarrow \infty . \end{aligned}$$

(4.16)

By (4.15) and (4.16) we find that there exists $T=T(\tau ,\omega ,D)>0$ such that for all $t\geq T$,

$$\begin{aligned} & \bigl\Vert u_{t}(\tau ,\tau -t,\theta _{-\tau}\omega ,u_{1,0}) \bigr\Vert ^{2}+\nu \bigl\Vert u( \tau ,\tau -t, \theta _{-\tau}\omega ,u_{0}) \bigr\Vert ^{2}+ \bigl\Vert \Delta u(\tau , \tau -t,\theta _{-\tau}\omega ,u_{0}) \bigr\Vert ^{2} \\ &\qquad {}+2 \int _{\mathbb{R}^{n}}F\bigl(x,u(\tau ,\tau -t,\theta _{-\tau} \omega ,u_{0})\bigr)\,dx \\ &\qquad {}+\varepsilon \bigl(u(\tau ,\tau -t,\theta _{-\tau}\omega ,u_{0}),u_{t}( \tau ,\tau -t,\theta _{-\tau}\omega ,u_{1,0})\bigr) \\ &\qquad {}+\biggl(\frac{1}{2}\alpha -\varepsilon -\frac{1}{8}\varepsilon \gamma \biggr) \int ^{\tau}_{\tau -t}e^{\frac{1}{4}\varepsilon \gamma (s-\tau )} \bigl\Vert u_{t}(s, \tau -t,\theta _{-\tau}\omega ,u_{1,0}) \bigr\Vert ^{2}\,ds \\ &\qquad {}+\varepsilon \biggl(1-\frac{1}{4}\gamma \biggr) \int ^{\tau}_{\tau -t}e^{ \frac{1}{4}\varepsilon \gamma (s-\tau )} \bigl\Vert \Delta u(s,\tau -t,\theta _{- \tau}\omega ,u_{0}) \bigr\Vert ^{2}\,ds \\ &\qquad {}+\frac{1}{2}\varepsilon \biggl(\nu -\frac{1}{2}\nu \gamma - \varepsilon \alpha +\frac{1}{4}\varepsilon ^{2}\gamma \biggr) \int ^{\tau}_{\tau -t}e^{ \frac{1}{4}\varepsilon \gamma (s-\tau )} \bigl\Vert u(s, \tau -t,\theta _{-\tau} \omega ,u_{0}) \bigr\Vert ^{2}\,ds \\ &\quad \leq 1+c_{5} \int ^{\tau}_{\tau -t}e^{\frac{1}{4}\varepsilon \gamma (s- \tau )} \bigl(1+ \bigl\Vert g(s) \bigr\Vert ^{2}+ \bigl\vert \zeta _{\delta}( \theta _{s-\tau}\omega ) \bigr\vert ^{2+ \frac{4}{p}} \bigr)\,ds. \end{aligned}$$

(4.17)

By (4.7) we obtain

$$\begin{aligned} &\varepsilon \bigl(u(\tau ,\tau -t,\theta _{-\tau}\omega ,u_{0}),u_{t}( \tau ,\tau -t,\theta _{-\tau}\omega ,u_{1,0})\bigr) \\ &\quad \leq\frac{1}{2}\varepsilon \bigl\Vert u(\tau ,\tau -t,\theta _{-\tau}\omega ,u_{0}) \bigr\Vert ^{2}+ \frac{1}{2}\varepsilon \bigl\Vert u_{t}(\tau ,\tau -t,\theta _{-\tau} \omega ,u_{1,0}) \bigr\Vert ^{2} \\ &\quad \leq\frac{1}{2}\nu \varepsilon \bigl\Vert u(\tau ,\tau -t,\theta _{-\tau} \omega ,u_{0}) \bigr\Vert ^{2}+ \frac{1}{2}\varepsilon \bigl\Vert u_{t}(\tau ,\tau -t, \theta _{-\tau}\omega ,u_{1,0}) \bigr\Vert ^{2}. \end{aligned}$$

(4.18)

It follows from (4.2), (4.17), and (4.18) that for all $t\geq T$,

$$\begin{aligned} &\frac{1}{2} \bigl\Vert u_{t}(\tau ,\tau -t,\theta _{-\tau}\omega ,u_{1,0}) \bigr\Vert ^{2}+ \frac{1}{2}\nu \bigl\Vert u(\tau ,\tau -t,\theta _{-\tau}\omega ,u_{0}) \bigr\Vert ^{2}+ \bigl\Vert \Delta u(\tau ,\tau -t,\theta _{-\tau}\omega ,u_{0}) \bigr\Vert ^{2} \\ &\qquad {}+\biggl(\frac{1}{2}\alpha -\varepsilon -\frac{1}{8}\varepsilon \gamma \biggr) \int ^{\tau}_{\tau -t}e^{\frac{1}{4}\varepsilon \gamma (s-\tau )} \bigl\Vert u_{t}(s, \tau -t,\theta _{-\tau}\omega ,u_{1,0}) \bigr\Vert ^{2}\,ds \\ &\qquad {}+\varepsilon \biggl(1-\frac{1}{4}\gamma \biggr) \int ^{\tau}_{\tau -t}e^{ \frac{1}{4}\varepsilon \gamma (s-\tau )} \bigl\Vert \Delta u(s,\tau -t,\theta _{- \tau}\omega ,u_{0}) \bigr\Vert ^{2}\,ds \\ &\qquad {}+\frac{1}{2}\varepsilon \biggl(\nu -\frac{1}{2}\nu \gamma - \varepsilon \alpha +\frac{1}{4}\varepsilon ^{2}\gamma \biggr) \int ^{\tau}_{\tau -t}e^{ \frac{1}{4}\varepsilon \gamma (s-\tau )} \bigl\Vert u(s, \tau -t,\theta _{-\tau} \omega ,u_{0}) \bigr\Vert ^{2}\,ds \\ &\quad \leq 1+2 \Vert \varphi _{1} \Vert _{L^{1}(\mathbb{R}^{n})}+c_{5} \int ^{0}_{- \infty}e^{\frac{1}{4}\varepsilon \gamma s} \bigl(1+ \bigl\Vert g(s+\tau ) \bigr\Vert ^{2}+ \bigl\vert \zeta _{\delta}(\theta _{s}\omega ) \bigr\vert ^{2+\frac{4}{p}} \bigr)\,ds. \end{aligned}$$

Then, the proof is completed. □

Based on Lemma 4.1, we can easily obtain the following Lemma that implies the existence of tempered random absorbing sets of Φ.

Lemma 4.2

If (3.2), (3.3), (3.6), (4.1), (4.2), and (4.5)–(4.9) hold, then the cocycle Φ possesses a closed measurable $\mathcal{D}$-pullback absorbing set $B=\{B(\tau ,\omega ):\tau \in \mathbb{R},\omega \in \Omega \}\in \mathcal{D}$, which is given by

$$ B(\tau ,\omega )=\bigl\{ (u_{0},u_{1,0})\in H^{2} \bigl(\mathbb{R}^{n}\bigr)\times L^{2}\bigl( \mathbb{R}^{n}\bigr): \Vert u_{0} \Vert ^{2}_{H^{2}(\mathbb{R}^{n})}+ \Vert u_{1,0} \Vert ^{2} \leq L(\tau ,\omega )\bigr\} , $$

(4.19)

where

$$ L(\tau ,\omega )=M_{1}+M_{1} \int ^{0}_{-\infty}e^{\frac{1}{4} \varepsilon \gamma s} \bigl(1+ \bigl\Vert g(s+\tau ) \bigr\Vert ^{2}+ \bigl\vert \zeta _{\delta}( \theta _{s}\omega ) \bigr\vert ^{2+\frac{4}{p}} \bigr)\,ds. $$

In order to derive the uniform tail-estimates of the solutions of (1.1) for large space variables when time is long enough, we need to derive the regularity of the solutions in a space higher than $H^{2}(\mathbb{R}^{n})$.

Lemma 4.3

Let (3.2), (3.3), (3.6), (4.1), (4.2), and (4.5)–(4.8) hold. Then, for any $\tau \in \mathbb{R}$, $\omega \in \Omega$ and $D\in \mathcal{D}$, there exists $T=T(\tau ,\omega ,D)>0$ such that for all $t\geq T$, the solution of (1.1) satisfies

$$\begin{aligned} & \bigl\Vert A^{\frac{1}{4}}u_{t}(\tau ,\tau -t,\theta _{-\tau}\omega ,u_{1,0}) \bigr\Vert ^{2}+ \bigl\Vert A^{\frac{3}{4}}u(\tau ,\tau -t,\theta _{-\tau}\omega ,u_{0}) \bigr\Vert ^{2} \\ &\qquad {}+ \int ^{\tau}_{\tau -t}e^{\frac{1}{4}\varepsilon \gamma (s-\tau )}\bigl( \bigl\Vert A^{\frac{1}{4}}u_{t}(s,\tau -t,\theta _{-\tau}\omega ,u_{1,0}) \bigr\Vert ^{2}+ \bigl\Vert A^{\frac{3}{4}}u(s, \tau -t,\theta _{-\tau}\omega ,u_{0}) \bigr\Vert ^{2}\bigr)\,ds \\ &\quad \leq M_{2}+M_{2} \int ^{0}_{-\infty}e^{\frac{1}{4}\varepsilon \gamma s}\bigl(1+ \bigl\Vert g(s+\tau ) \bigr\Vert ^{2}_{1}+ \bigl\vert \zeta _{\delta}(\theta _{s}\omega ) \bigr\vert ^{2} \bigr)\,ds, \end{aligned}$$

where $(u_{0},u_{1,0}) \in D(\tau -t,\theta _{-\tau}\omega )$ and $M_{2}$ is a positive number independent of τ, ω, and D.

Proof

Taking the inner product of (1.1)₁ with $A^{\frac{1}{2}} u$ in $L^{2}(\mathbb{R}^{n})$, we have

$$\begin{aligned} &\frac{d}{dt}\bigl(A^{\frac{1}{4}}u_{t},A^{\frac{1}{4}}u \bigr)+\alpha \bigl(A^{ \frac{1}{4}}u_{t},A^{\frac{1}{4}}u\bigr) + \bigl\Vert A^{\frac{3}{4}}u \bigr\Vert ^{2}+\nu \bigl\Vert A^{ \frac{1}{4}}u \bigr\Vert ^{2}+\bigl(f(x,u),A^{\frac{1}{2}}u \bigr) \\ &\quad = \bigl\Vert A^{\frac{1}{4}}u_{t} \bigr\Vert ^{2}+ \bigl(g(t)+h(t,\cdot ,u)\zeta _{\delta}( \theta _{t}\omega ),A^{\frac{1}{2}} u\bigr). \end{aligned}$$

(4.20)

Taking the inner product of (1.1)₁ with $A^{\frac{1}{2}} u_{t}$ in $L^{2}(\mathbb{R}^{n})$, we find that

$$\begin{aligned} &\frac{d}{dt}\bigl( \bigl\Vert A^{\frac{1}{4}}u_{t} \bigr\Vert ^{2}+\nu \bigl\Vert A^{\frac{1}{4}}u \bigr\Vert ^{2}+ \bigl\Vert A^{\frac{3}{4}} u \bigr\Vert ^{2}\bigr) \\ &\quad =-2\alpha \bigl\Vert A^{\frac{1}{4}}u_{t} \bigr\Vert ^{2}-2\bigl(f(x,u),A^{\frac{1}{2}} u_{t}\bigr)+2 \bigl(g(t)+h(t, \cdot ,u)\zeta _{\delta}(\theta _{t}\omega ),A^{\frac{1}{2}} u_{t}\bigr). \end{aligned}$$

(4.21)

By (4.20) and (4.21), we obtain

$$\begin{aligned} &\frac{d}{dt} \bigl( \bigl\Vert A^{\frac{1}{4}}u_{t} \bigr\Vert ^{2}+\nu \bigl\Vert A^{\frac{1}{4}}u \bigr\Vert ^{2}+ \bigl\Vert A^{\frac{3}{4}} u \bigr\Vert ^{2}+ \varepsilon \bigl(A^{\frac{1}{4}}u_{t},A^{ \frac{1}{4}}u\bigr) \bigr) \\ &\qquad {}+(2\alpha -\varepsilon ) \bigl\Vert A^{\frac{1}{4}}u_{t} \bigr\Vert ^{2} +\varepsilon \alpha \bigl(A^{\frac{1}{4}}u_{t},A^{\frac{1}{4}}u \bigr) \\ &\qquad {}+\varepsilon \bigl\Vert A^{\frac{3}{4}}u \bigr\Vert ^{2}+ \varepsilon \nu \bigl\Vert A^{ \frac{1}{4}}u \bigr\Vert ^{2}+ \varepsilon \bigl(f(x,u),A^{\frac{1}{2}} u\bigr)+2\bigl(f(x,u),A^{ \frac{1}{2}} u_{t}\bigr) \\ &\quad =\bigl(g(t)+h(t,\cdot ,u)\zeta _{\delta}(\theta _{t}\omega ), \varepsilon A^{ \frac{1}{2}} u+2A^{\frac{1}{2}} u_{t}\bigr). \end{aligned}$$

(4.22)

For the right-hand side of (4.22), using (4.5), (4.6), and Lemma 4.1, we have

$$\begin{aligned} &\bigl(g(t)+h\bigl(t,\cdot ,u(t)\bigr)\zeta _{\delta}(\theta _{t}\omega ), \varepsilon A^{\frac{1}{2}}u+2A^{\frac{1}{2}}u_{t} \bigr) \\ &\quad \leq\bigl( \bigl\Vert g(t) \bigr\Vert _{1}+ \bigl\Vert h \bigl(t,\cdot ,u(t)\bigr)\zeta _{\delta}(\theta _{t} \omega ) \bigr\Vert _{1}\bigr) \bigl(\varepsilon \bigl\Vert A^{\frac{1}{4}} u \bigr\Vert +2 \bigl\Vert A^{\frac{1}{2}}u_{t} \bigr\Vert \bigr) \\ &\quad \leq\frac{1}{2}\varepsilon \nu \bigl\Vert A^{\frac{1}{4}}u \bigr\Vert ^{2}+\alpha \bigl\Vert A^{ \frac{1}{2}}u_{t} \bigr\Vert ^{2}+\biggl(\alpha ^{-1}+\frac{1}{2}\varepsilon \nu ^{-1}\biggr) \bigl( \bigl\Vert g(t) \bigr\Vert _{1}+ \bigl\Vert h\bigl(t,\cdot ,u(t)\bigr)\zeta _{\delta}(\theta _{t}\omega ) \bigr\Vert _{1}\bigr)^{2} \\ &\quad \leq\frac{1}{2}\varepsilon \nu \bigl\Vert A^{\frac{1}{4}}u \bigr\Vert ^{2}+\alpha \bigl\Vert A^{ \frac{1}{4}}u_{t} \bigr\Vert ^{2}+\bigl(2\alpha ^{-1} +\varepsilon \nu ^{-1}\bigr) \bigl\Vert g(t) \bigr\Vert ^{2}_{1} \\ &\qquad {}+ \bigl(2\alpha ^{-1}+\varepsilon \nu ^{-1}\bigr) \bigl\Vert h \bigl(t,\cdot ,u(t)\bigr) \zeta _{\delta}(\theta _{t}\omega ) \bigr\Vert ^{2}_{1} \\ &\quad \leq\frac{1}{2}\varepsilon \nu \bigl\Vert A^{\frac{1}{4}}u \bigr\Vert ^{2}+\alpha \bigl\Vert A^{ \frac{1}{4}}u_{t} \bigr\Vert ^{2}+\bigl(2\alpha ^{-1} +\varepsilon \nu ^{-1}\bigr) \bigl\Vert g(t) \bigr\Vert ^{2}_{1}+c_{8} \bigl\vert \zeta _{\delta}(\theta _{t}\omega ) \bigr\vert ^{2}. \end{aligned}$$

(4.23)

From (4.3) and Lemma 4.1 we find

$$\begin{aligned} & \bigl\vert \varepsilon \bigl(f(x,u),A^{\frac{1}{2}} u\bigr)+2 \bigl(f(x,u),A^{\frac{1}{2}} u_{t}\bigr) \bigr\vert \\ &\quad \leq2 \int _{\mathbb{R}^{n}} \biggl\vert \frac{\partial f}{\partial u}(x,u) \cdot A^{\frac{1}{4}} u\cdot A^{\frac{1}{4}} u_{t}+ \frac{\partial f}{\partial x}(x,u) \cdot A^{\frac{1}{4}} u_{t} \biggr\vert \,dx \\ &\qquad {}+\varepsilon \int _{\mathbb{R}^{n}} \biggl\vert \frac{\partial f}{\partial u}(x,u) \cdot A^{\frac{1}{4}} u\cdot A^{\frac{1}{4}} u+ \frac{\partial f}{\partial x}(x,u)\cdot A^{\frac{1}{4}} u \biggr\vert \,dx \\ &\quad \leq2\iota \int _{\mathbb{R}^{n}} \vert u \vert ^{p}\cdot \bigl\vert A^{\frac{1}{4}} u \bigr\vert \cdot \bigl\vert A^{\frac{1}{4}} u_{t} \bigr\vert \,dx+2\varsigma \int _{\mathbb{R}^{n}} \bigl\vert A^{ \frac{1}{4}} u \bigr\vert \cdot \bigl\vert A^{\frac{1}{4}} u_{t} \bigr\vert \,dx \\ &\qquad {}+2 \int _{\mathbb{R}^{n}} \vert \varphi _{4} \vert \cdot \bigl\vert A^{\frac{1}{4}} u_{t} \bigr\vert \,dx \\ &\qquad {}+\varepsilon \iota \int _{\mathbb{R}^{n}} \vert u \vert ^{p}\cdot \bigl\vert A^{ \frac{1}{4}} u \bigr\vert \cdot \bigl\vert A^{\frac{1}{4}} u \bigr\vert \,dx+\varepsilon \varsigma \int _{\mathbb{R}^{n}} \bigl\vert A^{\frac{1}{4}} u \bigr\vert \cdot \bigl\vert A^{\frac{1}{4}} u \bigr\vert \,dx \\ &\qquad {}+ \varepsilon \int _{\mathbb{R}^{n}} \vert \varphi _{4} \vert \cdot \bigl\vert A^{\frac{1}{4}} u \bigr\vert \,dx \\ &\quad \leq2\iota \Vert u \Vert ^{p}_{L^{\frac{10p}{4}}}\cdot \bigl\Vert A^{\frac{1}{4}} u \bigr\Vert _{L^{10}} \cdot \bigl\Vert A^{\frac{1}{4}} u_{t} \bigr\Vert +2\varsigma \bigl\Vert A^{\frac{1}{4}} u \bigr\Vert \cdot \bigl\Vert A^{\frac{1}{4}} u_{t} \bigr\Vert +\frac{\varepsilon}{4} \bigl\Vert A^{\frac{1}{4}} u_{t} \bigr\Vert ^{2}+ \frac{4}{\varepsilon} \Vert \varphi _{4} \Vert ^{2} \\ &\qquad {}+\varepsilon \iota \Vert u \Vert ^{p}\cdot \bigl\Vert A^{\frac{1}{4}} u \bigr\Vert ^{2} + \varepsilon \varsigma \bigl\Vert A^{\frac{1}{4}} u \bigr\Vert ^{2}+\frac{\varepsilon}{2} \bigl\Vert A^{\frac{1}{4}} u \bigr\Vert ^{2}+ \frac{\varepsilon}{2} \Vert \varphi _{4} \Vert ^{2} \\ &\quad \leq\varepsilon \bigl\Vert A^{\frac{1}{4}} u_{t} \bigr\Vert ^{2}+ \frac{2C^{p+1}\iota ^{2}}{\varepsilon}L^{p} \bigl\Vert A^{\frac{3}{4}} u \bigr\Vert ^{2}+c_{9}, \end{aligned}$$

where the definition of L see Lemma 4.2, and C is the positive constant satisfying

$$ C \Vert \Delta u \Vert ^{2}\geq \biggl( \int _{\mathbb{R}^{n}} \vert u \vert ^{10}\,dx \biggr)^{ \frac{1}{5}}, \qquad C \Vert u \Vert ^{2}_{2}\geq \biggl( \int _{\mathbb{R}^{n}} \vert u \vert ^{\frac{10p}{4}}\,dx \biggr)^{\frac{2}{10p}}. $$

Choosing

$$ 0< \iota ^{2}\leq \frac{\varepsilon ^{2}}{4L^{p}C^{p+1}}, $$

we obtain

$$ \bigl\vert \varepsilon \bigl(f(x,u),A^{\frac{1}{2}} u\bigr)+2 \bigl(f(x,u),A^{\frac{1}{2}} u_{t}\bigr) \bigr\vert \leq \varepsilon \bigl\Vert A^{\frac{1}{4}} u_{t} \bigr\Vert ^{2}+ \frac{\varepsilon}{2} \bigl\Vert A^{\frac{3}{4}} u \bigr\Vert ^{2}+c_{9}. $$

(4.24)

By (4.22)–(4.24), we obtain

$$\begin{aligned} &\frac{d}{dt} \bigl( \bigl\Vert A^{\frac{1}{4}}u_{t} \bigr\Vert ^{2}+\nu \bigl\Vert A^{\frac{1}{4}}u \bigr\Vert ^{2}+ \bigl\Vert A^{\frac{3}{4}} u \bigr\Vert ^{2}+ \varepsilon \bigl(A^{\frac{1}{4}}u_{t},A^{ \frac{1}{4}}u\bigr) \bigr)+(\alpha -2\varepsilon ) \bigl\Vert A^{\frac{1}{4}}u_{t} \bigr\Vert ^{2} \\ &\qquad {}+\varepsilon \alpha \bigl(A^{\frac{1}{4}}u_{t},A^{\frac{1}{4}}u \bigr)+ \frac{\varepsilon}{2} \bigl\Vert A^{\frac{3}{4}}u \bigr\Vert ^{2}+\frac{\varepsilon}{2} \nu \bigl\Vert A^{\frac{1}{4}}u \bigr\Vert ^{2} \\ &\quad \leq c_{10}\bigl(1+ \bigl\Vert g(t) \bigr\Vert ^{2}_{1}+ \bigl\vert \zeta _{\delta}(\theta _{t}\omega ) \bigr\vert ^{2}\bigr), \end{aligned}$$

which can be rewritten as

$$\begin{aligned} &\frac{d}{dt} \bigl( \bigl\Vert A^{\frac{1}{4}}u_{t} \bigr\Vert ^{2}+\nu \bigl\Vert A^{\frac{1}{4}}u \bigr\Vert ^{2}+ \bigl\Vert A^{\frac{3}{4}} u \bigr\Vert ^{2}+ \varepsilon \bigl(A^{\frac{1}{4}}u_{t},A^{ \frac{1}{4}}u\bigr) \bigr) \\ &\qquad {}+\frac{1}{4}\varepsilon \gamma \bigl( \bigl\Vert A^{\frac{1}{4}}u_{t} \bigr\Vert ^{2}+ \nu \bigl\Vert A^{\frac{1}{4}}u \bigr\Vert ^{2}+ \bigl\Vert A^{\frac{3}{4}} u \bigr\Vert ^{2}+ \varepsilon \bigl(A^{ \frac{1}{4}}u_{t},A^{\frac{1}{4}}u\bigr) \bigr) \\ &\qquad {}+\biggl(\alpha -2\varepsilon -\frac{1}{4}\varepsilon \gamma \biggr) \bigl\Vert A^{ \frac{1}{4}}u_{t} \bigr\Vert ^{2} + \frac{\varepsilon}{2}\biggl(1-\frac{\gamma}{2}\biggr) \bigl\Vert A^{ \frac{3}{4}}u \bigr\Vert ^{2} +\frac{\varepsilon}{2}\nu \biggl(1- \frac{\gamma}{2}\biggr) \bigl\Vert A^{ \frac{1}{4}}u \bigr\Vert ^{2} \\ &\quad \leq c_{10}\bigl(1+ \bigl\Vert g(t) \bigr\Vert ^{2}_{1}+ \bigl\vert \zeta _{\delta}(\theta _{t}\omega ) \bigr\vert ^{2}\bigr) -\varepsilon \biggl( \alpha -\frac{1}{4}\varepsilon \gamma \biggr) \bigl(A^{\frac{1}{4}}u_{t},A^{ \frac{1}{4}}u \bigr). \end{aligned}$$

(4.25)

For the last term on the right-hand side of (4.25) we have

$$\begin{aligned} &-\varepsilon \biggl(\alpha -\frac{1}{4}\varepsilon \gamma \biggr) \bigl(A^{\frac{1}{4}}u_{t},A^{ \frac{1}{4}}u\bigr) \\ &\quad \leq\varepsilon \biggl(\alpha -\frac{1}{4}\varepsilon \gamma \biggr) \bigl\Vert A^{ \frac{1}{4}}u \bigr\Vert \bigl\Vert A^{\frac{1}{4}}u_{t} \bigr\Vert \\ &\quad \leq\frac{1}{2}\varepsilon ^{2}\biggl(\alpha - \frac{1}{4}\varepsilon \gamma \biggr) \bigl\Vert A^{\frac{1}{4}}u \bigr\Vert ^{2}+ \frac{1}{2}\biggl(\alpha -\frac{1}{4} \varepsilon \gamma \biggr) \bigl\Vert A^{\frac{1}{4}}u_{t} \bigr\Vert ^{2}, \end{aligned}$$

from which together with (4.25), we obtain

$$\begin{aligned} &\frac{d}{dt} \bigl( \bigl\Vert A^{\frac{1}{4}}u_{t} \bigr\Vert ^{2}+\nu \bigl\Vert A^{\frac{1}{4}}u \bigr\Vert ^{2}+ \bigl\Vert A^{\frac{3}{4}} u \bigr\Vert ^{2}+ \varepsilon \bigl(A^{\frac{1}{4}}u_{t},A^{ \frac{1}{4}}u\bigr) \bigr) \\ &\qquad {}+\frac{1}{4}\varepsilon \gamma \bigl( \bigl\Vert A^{\frac{1}{4}}u_{t} \bigr\Vert ^{2}+ \nu \bigl\Vert A^{\frac{1}{4}}u \bigr\Vert ^{2}+ \bigl\Vert A^{\frac{3}{4}} u \bigr\Vert ^{2}+ \varepsilon \bigl(A^{ \frac{1}{4}}u_{t},A^{\frac{1}{4}}u\bigr) \bigr) \\ &\qquad {}+\biggl(\frac{\alpha}{2}-2\varepsilon -\frac{1}{8}\varepsilon \gamma \biggr) \bigl\Vert A^{ \frac{1}{4}}u_{t} \bigr\Vert ^{2} +\frac{\varepsilon}{2}\biggl(1-\frac{\gamma}{2}\biggr) \bigl\Vert A^{ \frac{3}{4}}u \bigr\Vert ^{2} \\ &\qquad {} +\frac{\varepsilon}{2}\biggl(\nu - \frac{\nu}{2}\gamma - \frac{\varepsilon}{2}\alpha +\frac{1}{8}\varepsilon ^{2}\gamma \biggr) \bigl\Vert A^{ \frac{1}{4}}u \bigr\Vert ^{2} \\ &\quad \leq c_{10}\bigl(1+ \bigl\Vert g(t) \bigr\Vert ^{2}_{1}+ \bigl\vert \zeta _{\delta}(\theta _{t}\omega ) \bigr\vert ^{2}\bigr). \end{aligned}$$

(4.26)

Similar to the remainder of Lemma 4.1, we can obtain the desired result. □

Lemma 4.4

Let (3.2), (3.3), (3.6), (4.1), (4.2), and (4.5)–(4.8) hold. Then, for every $\eta >0$, $\tau \in \mathbb{R}$, $\omega \in \Omega$ and $D\in \mathcal{D}$, there exists $T_{0}=T_{0}(\eta ,\tau ,\omega ,D)>0$ and $m_{0}=m_{0}(\eta ,\tau ,\omega )\geq 1$ such that for all $t\geq T_{0}$, $m\geq m_{0}$ and $(u_{0},u_{1,0}) \in D(\tau -t,\theta _{-\tau}\omega )$, the solution of (1.1) satisfies

$$\begin{aligned} &\int _{ \vert x \vert \geq m}\bigl( \bigl\vert u_{t}(\tau ,\tau -t, \theta _{-\tau}\omega ,u_{1,0}) \bigr\vert ^{2}+ \bigl\vert u( \tau ,\tau -t,\theta _{-\tau}\omega ,u_{0}) \bigr\vert ^{2} \\ &\quad {}+ \bigl\vert \Delta u(\tau , \tau -t,\theta _{-\tau}\omega ,u_{0}) \bigr\vert ^{2}\bigr)\,dx< \eta . \end{aligned}$$

Proof

Let $\rho :\mathbb{R}^{n}\rightarrow \mathbb{R}$ be a smooth function such that $0\leq \rho (x)\leq 1$ for all $x\in \mathbb{R}^{n}$, and

$$ \rho (x)=0 \quad \text{for } \vert x \vert \leq \frac{1}{2};\quad \text{and}\quad \rho (x)=1 \quad \text{for } \vert x \vert \geq 1. $$

For every $m\in \mathbb{N}$, let

$$ \rho _{m}(x)=\rho (x/m),\quad x\in \mathbb{R}^{n}. $$

Then, there exist positive constants $c_{11}$ and $c_{12}$ independent of m such that $|\nabla \rho _{m}(x)|\leq \frac{1}{m}c_{11}$, $|\Delta \rho _{m}(x)|\leq \frac{1}{m}c_{12}$ for all $x\in \mathbb{R}^{n}$ and $m\in \mathbb{N}$.

Similar to the energy equation (3.11), we have

$$\begin{aligned} &\frac{d}{dt} \int _{\mathbb{R}^{n}}\rho _{m}(x) \bigl( \bigl\vert u_{t}(t,x) \bigr\vert ^{2}+ \nu \bigl\vert u(t,x) \bigr\vert ^{2}+ \bigl\vert \Delta u(t,x) \bigr\vert ^{2}+2F \bigl(x,u(t,x)\bigr) \bigr)\,dx \\ &\qquad {}+2\alpha \int _{\mathbb{R}^{n}}\rho _{m}(x) \bigl\vert u_{t}(t,x) \bigr\vert ^{2}\,dx \\ &\quad =-4 \int _{\mathbb{R}^{n}}\nabla \rho _{m}(x)\cdot \Delta u(t,x) \cdot \nabla u_{t}(t,x)\,dx-2 \int _{\mathbb{R}^{n}}\Delta \rho _{m}(x) \cdot \Delta u(t,x) \cdot u_{t}(t,x)\,dx \\ &\qquad {}+2 \int _{\mathbb{R}^{n}}\rho _{m}(x) g(t,x) u_{t}(t,x)\,dx \\ &\qquad {}+2\zeta _{ \delta}(\theta _{t}\omega ) \int _{\mathbb{R}^{n}}\rho _{m}(x)h\bigl(t,x,u(t,x) \bigr)u_{t}(t,x)\,dx. \end{aligned}$$

(4.27)

Taking the inner product of (1.1)₁ with $\rho _{m}(x)u$ in $L^{2}(\mathbb{R}^{n})$, we have

$$\begin{aligned} &\frac{d}{dt} \int _{\mathbb{R}^{n}}\rho _{m}(x)u(t,x)u_{t}(t,x)\,dx+ \alpha \int _{\mathbb{R}^{n}}\rho _{m}(x)u(t,x)u_{t}(t,x)\,dx \\ &\qquad {}+ \int _{\mathbb{R}^{n}}\rho _{m}(x) \bigl\vert \Delta u(t,x) \bigr\vert ^{2}\,dx+\nu \int _{ \mathbb{R}^{n}}\rho _{m}(x) \bigl\vert u(t,x) \bigr\vert ^{2}\,dx \\ &\qquad {}+ \int _{\mathbb{R}^{n}}\rho _{m}(x)f\bigl(x,u(t,x)\bigr)u(t,x)\,dx \\ &\quad = \int _{\mathbb{R}^{n}}\rho _{m}(x) \bigl\vert u_{t}(t,x) \bigr\vert ^{2}\,dx-2 \int _{ \mathbb{R}^{n}}\nabla \rho _{m}(x)\cdot \Delta u(t,x) \cdot \nabla u(t,x)\,dx \\ &\qquad {}- \int _{\mathbb{R}^{n}}\Delta \rho _{m}(x)\cdot \Delta u(t,x) \cdot u(t,x)\,dx \\ &\qquad {}+ \int _{\mathbb{R}^{n}}\rho _{m}(x) \bigl(g(t,x)+h \bigl(t,x,u(t,x)\bigr)\zeta _{ \delta}(\theta _{t}\omega ) \bigr)u(t,x)\,dx. \end{aligned}$$

(4.28)

By (4.27) and (4.28), we obtain

$$\begin{aligned} &\frac{d}{dt} \int _{\mathbb{R}^{n}}\rho _{m}(x) \bigl( \bigl\vert u_{t}(t,x) \bigr\vert ^{2}+ \nu \bigl\vert u(t,x) \bigr\vert ^{2}+ \bigl\vert \Delta u(t,x) \bigr\vert ^{2} \\ &\qquad {}+2F \bigl(x,u(t,x)\bigr)+\varepsilon u(t,x)u_{t}(t,x) \bigr)\,dx \\ &\qquad {}+(2\alpha -\varepsilon ) \int _{\mathbb{R}^{n}}\rho _{m}(x) \bigl\vert u_{t}(t,x) \bigr\vert ^{2}\,dx +\varepsilon \alpha \int _{\mathbb{R}^{n}}\rho _{m}(x)u(t,x)u_{t}(t,x)\,dx \\ &\qquad {}+ \varepsilon \int _{\mathbb{R}^{n}}\rho _{m}(x) \bigl\vert \Delta u(t,x) \bigr\vert ^{2}\,dx \\ &\qquad {}+\varepsilon \nu \int _{\mathbb{R}^{n}}\rho _{m}(x) \bigl\vert u(t,x) \bigr\vert ^{2}\,dx+ \varepsilon \int _{\mathbb{R}^{n}}\rho _{m}(x)f\bigl(x,u(t,x)\bigr)u(t,x)\,dx \\ &\quad = \int _{\mathbb{R}^{n}}\rho _{m}(x) \bigl(g(t,x)+h \bigl(t,x,u(t,x)\bigr)\zeta _{ \delta}(\theta _{t}\omega )\bigr) \bigl(\varepsilon u(t,x)+2u_{t}(t,x)\bigr)\,dx \\ &\qquad {}-2\varepsilon \int _{\mathbb{R}^{n}}\nabla \rho _{m}(x)\cdot \Delta u(t,x) \cdot \nabla u(t,x)\,dx-\varepsilon \int _{\mathbb{R}^{n}}\Delta \rho _{m}(x) \cdot \Delta u(t,x) \cdot u(t,x)\,dx \\ &\qquad {}-4 \int _{\mathbb{R}^{n}}\nabla \rho _{m}(x)\cdot \Delta u(t,x) \cdot \nabla u_{t}(t,x)\,dx \\ &\qquad {}-2 \int _{\mathbb{R}^{n}}\Delta \rho _{m}(x)\cdot \Delta u(t,x) \cdot u_{t}(t,x)\,dx. \end{aligned}$$

(4.29)

By (4.1) and (4.29) we obtain

$$\begin{aligned} &\frac{d}{dt} \int _{\mathbb{R}^{n}}\rho _{m}(x) \bigl( \bigl\vert u_{t}(t,x) \bigr\vert ^{2}+ \nu \bigl\vert u(t,x) \bigr\vert ^{2}+ \bigl\vert \Delta u(t,x) \bigr\vert ^{2} \\ &\qquad {}+2F \bigl(x,u(t,x)\bigr)+\varepsilon u(t,x)u_{t}(t,x) \bigr)\,dx \\ &\qquad {}+(2\alpha -\varepsilon ) \int _{\mathbb{R}^{n}}\rho _{m}(x) \bigl\vert u_{t}(t,x) \bigr\vert ^{2}\,dx +\varepsilon \alpha \int _{\mathbb{R}^{n}}\rho _{m}(x)u(t,x)u_{t}(t,x)\,dx \\ &\qquad {}+ \varepsilon \int _{\mathbb{R}^{n}}\rho _{m}(x) \bigl\vert \Delta u(t,x) \bigr\vert ^{2}\,dx \\ &\qquad {}+\varepsilon \nu \int _{\mathbb{R}^{n}}\rho _{m}(x) \bigl\vert u(t,x) \bigr\vert ^{2}\,dx+ \varepsilon \gamma \int _{\mathbb{R}^{n}}\rho _{m}(x)F\bigl(x,u(t,x)\bigr)\,dx \\ &\quad \leq \int _{\mathbb{R}^{n}}\rho _{m}(x) \bigl(g(t,x)+h \bigl(t,x,u(t,x)\bigr)\zeta _{ \delta}(\theta _{t}\omega )\bigr) \bigl(\varepsilon u(t,x)+2u_{t}(t,x)\bigr)\,dx \\ &\qquad {}- \varepsilon \int _{\mathbb{R}^{n}}\rho _{m}(x)\varphi _{3}(x)\,dx \\ &\qquad {}-2\varepsilon \int _{\mathbb{R}^{n}}\nabla \rho _{m}(x)\cdot \Delta u(t,x) \cdot \nabla u(t,x)\,dx-\varepsilon \int _{\mathbb{R}^{n}}\Delta \rho _{m}(x) \cdot \Delta u(t,x) \cdot u(t,x)\,dx \\ &\qquad {}-4 \int _{\mathbb{R}^{n}}\nabla \rho _{m}(x)\cdot \Delta u(t,x) \cdot \nabla u_{t}(t,x)\,dx \\ &\qquad {}-2 \int _{\mathbb{R}^{n}}\Delta \rho _{m}(x)\cdot \Delta u(t,x) \cdot u_{t}(t,x)\,dx. \end{aligned}$$

(4.30)

Similar to the arguments of (4.11), we know that the first term on the right-hand side of (4.30) is bounded by

$$\begin{aligned} & \biggl\vert \int _{\mathbb{R}^{n}}\rho _{m}(x) \bigl(g(t,x)+h \bigl(t,x,u(t,x)\bigr)\zeta _{ \delta}(\theta _{t}\omega )\bigr) \bigl(\varepsilon u(t,x)+2u_{t}(t,x)\bigr)\,dx \biggr\vert \\ &\quad \leq\frac{1}{2}\varepsilon \nu \int _{\mathbb{R}^{n}}\rho _{m}(x) \bigl\vert u(t,x) \bigr\vert ^{2}\,dx +\alpha \int _{\mathbb{R}^{n}}\rho _{m}(x) \bigl\vert u_{t}(t,x) \bigr\vert ^{2}\,dx \\ &\qquad {}+ \frac{1}{2} \varepsilon \gamma \int _{\mathbb{R}^{n}}\rho _{m}(x)F\bigl(x,u(t,x)\bigr)\,dx \\ &\qquad {}+c_{13} \int _{\mathbb{R}^{n}}\rho _{m}(x) \bigl( \bigl\vert g(t,x) \bigr\vert ^{2}+ \bigl\vert \varphi _{1}(x) \bigr\vert + \bigl\vert \zeta _{\delta}(\theta _{t}\omega )\varphi _{6}(x) \bigr\vert ^{2} \\ &\qquad {}+ \bigl\vert \zeta _{\delta}(\theta _{t}\omega )\varphi _{5}(x) \bigr\vert ^{2+\frac{4}{p}} \bigr)\,dx, \end{aligned}$$

(4.31)

where $c_{13}$ depends only on α, ν, γ, and ε.

By (4.30) and (4.31) we obtain

$$\begin{aligned} &\frac{d}{dt} \int _{\mathbb{R}^{n}}\rho _{m}(x) \bigl( \bigl\vert u_{t}(t,x) \bigr\vert ^{2}+ \nu \bigl\vert u(t,x) \bigr\vert ^{2}+ \bigl\vert \Delta u(t,x) \bigr\vert ^{2} \\ &\qquad {}+2F \bigl(x,u(t,x)\bigr)+\varepsilon u(t,x)u_{t}(t,x) \bigr)\,dx \\ &\qquad {}+(\alpha -\varepsilon ) \int _{\mathbb{R}^{n}}\rho _{m}(x) \bigl\vert u_{t}(t,x) \bigr\vert ^{2}\,dx +\varepsilon \alpha \int _{\mathbb{R}^{n}}\rho _{m}(x)u(t,x)u_{t}(t,x)\,dx \\ &\qquad {}+ \varepsilon \int _{\mathbb{R}^{n}}\rho _{m}(x) \bigl\vert \Delta u(t,x) \bigr\vert ^{2}\,dx \\ &\qquad {}+\frac{1}{2}\varepsilon \nu \int _{\mathbb{R}^{n}}\rho _{m}(x) \bigl\vert u(t,x) \bigr\vert ^{2}\,dx+ \frac{1}{2}\varepsilon \gamma \int _{\mathbb{R}^{n}}\rho _{m}(x)F\bigl(x,u(t,x)\bigr)\,dx \\ &\quad \leq c_{14} \int _{\mathbb{R}^{n}}\rho _{m}(x) \bigl( \bigl\vert g(t,x) \bigr\vert ^{2}+ \bigl\vert \varphi _{1}(x) \bigr\vert + \bigl\vert \varphi _{3}(x) \bigr\vert + \bigl\vert \zeta _{\delta}(\theta _{t} \omega )\varphi _{6}(x) \bigr\vert ^{2} \\ &\qquad {} + \bigl\vert \zeta _{\delta}(\theta _{t} \omega ) \varphi _{5}(x) \bigr\vert ^{2+\frac{4}{p}} \bigr)\,dx \\ &\qquad {}+\frac{c_{14}}{m}\bigl( \Vert u \Vert + \Vert \nabla u \Vert + \Vert u_{t} \Vert + \Vert \nabla u_{t} \Vert \bigr) \Vert \Delta u \Vert , \end{aligned}$$

(4.32)

where $c_{14}>0$ depends only on α, ν, γ, and ε, but not on m.

By (4.32) we obtain

$$\begin{aligned} &\frac{d}{dt} \int _{\mathbb{R}^{n}}\rho _{m}(x) \bigl( \bigl\vert u_{t}(t,x) \bigr\vert ^{2}+ \nu \bigl\vert u(t,x) \bigr\vert ^{2}+ \bigl\vert \Delta u(t,x) \bigr\vert ^{2} \\ &\qquad {}+2F \bigl(x,u(t,x)\bigr)+\varepsilon u(t,x)u_{t}(t,x) \bigr)\,dx \\ &\qquad {}+\frac{1}{4}\varepsilon \gamma \int _{\mathbb{R}^{n}}\rho _{m}(x) \bigl( \bigl\vert u_{t}(t,x) \bigr\vert ^{2}+\nu \bigl\vert u(t,x) \bigr\vert ^{2}+ \bigl\vert \Delta u(t,x) \bigr\vert ^{2} \\ &\qquad {}+2F \bigl(x,u(t,x)\bigr)+ \varepsilon u(t,x)u_{t}(t,x) \bigr)\,dx \\ &\qquad {}+\biggl(\alpha -\varepsilon -\frac{1}{4}\varepsilon \gamma \biggr) \int _{ \mathbb{R}^{n}}\rho _{m}(x) \bigl\vert u_{t}(t,x) \bigr\vert ^{2}\,dx \\ &\qquad {}+\varepsilon \biggl(\alpha - \frac{1}{4}\gamma \biggr) \int _{\mathbb{R}^{n}}\rho _{m}(x)u(t,x)u_{t}(t,x)\,dx \\ &\qquad {}+\varepsilon \biggl(1-\frac{1}{4}\gamma \biggr) \int _{\mathbb{R}^{n}}\rho _{m}(x) \bigl\vert \Delta u(t,x) \bigr\vert ^{2}\,dx \\ &\qquad {}+\frac{1}{2}\varepsilon \nu \biggl(1- \frac{1}{2} \gamma \biggr) \int _{\mathbb{R}^{n}}\rho _{m}(x) \bigl\vert u(t,x) \bigr\vert ^{2}\,dx+\frac{1}{2} \varepsilon \gamma \int _{\mathbb{R}^{n}}\rho _{m}(x)F\bigl(x,u(t,x)\bigr)\,dx \\ &\quad \leq c_{14} \int _{\mathbb{R}^{n}}\rho _{m}(x) \bigl( \bigl\vert g(t,x) \bigr\vert ^{2}+ \bigl\vert \varphi _{1}(x) \bigr\vert + \bigl\vert \varphi _{3}(x) \bigr\vert + \bigl\vert \zeta _{\delta}(\theta _{t} \omega )\varphi _{6}(x) \bigr\vert ^{2} \\ &\qquad {} + \bigl\vert \zeta _{\delta}(\theta _{t} \omega ) \varphi _{5}(x) \bigr\vert ^{2+\frac{4}{p}} \bigr)\,dx \\ &\qquad {}+\frac{c_{14}}{m}\bigl( \Vert u \Vert + \Vert \nabla u \Vert + \Vert u_{t} \Vert + \Vert \nabla u_{t} \Vert \bigr) \Vert \Delta u \Vert . \end{aligned}$$

(4.33)

By Young’s inequality we obtain

$$\begin{aligned} & \biggl\vert \varepsilon \biggl(\alpha -\frac{1}{4}\gamma \biggr) \int _{\mathbb{R}^{n}} \rho _{m}(x)u(t,x)u_{t}(t,x)\,dx \biggr\vert \\ &\quad \leq\frac{1}{2}\varepsilon ^{2}\biggl(\alpha - \frac{1}{4}\gamma \biggr) \int _{ \mathbb{R}^{n}}\rho _{m}(x) \bigl\vert u(t,x) \bigr\vert ^{2}\,dx \\ &\qquad {}+ \frac{1}{2}\biggl(\alpha - \frac{1}{4} \gamma \biggr) \int _{\mathbb{R}^{n}}\rho _{m}(x) \bigl\vert u_{t}(t,x) \bigr\vert ^{2}\,dx. \end{aligned}$$

(4.34)

By (4.33) and (4.34) we obtain

$$\begin{aligned} &\frac{d}{dt} \int _{\mathbb{R}^{n}}\rho _{m}(x) \bigl( \bigl\vert u_{t}(t,x) \bigr\vert ^{2}+ \nu \bigl\vert u(t,x) \bigr\vert ^{2}+ \bigl\vert \Delta u(t,x) \bigr\vert ^{2} \\ &\qquad {}+2F \bigl(x,u(t,x)\bigr)+\varepsilon u(t,x)u_{t}(t,x) \bigr)\,dx \\ &\qquad {}+\frac{1}{4}\varepsilon \gamma \int _{\mathbb{R}^{n}}\rho _{m}(x) \bigl( \bigl\vert u_{t}(t,x) \bigr\vert ^{2}+\nu \bigl\vert u(t,x) \bigr\vert ^{2}+ \bigl\vert \Delta u(t,x) \bigr\vert ^{2} \\ &\qquad {}+2F \bigl(x,u(t,x)\bigr)+ \varepsilon u(t,x)u_{t}(t,x) \bigr)\,dx \\ &\qquad {}+\biggl(\frac{1}{2}\alpha -\varepsilon -\frac{1}{8}\varepsilon \gamma \biggr) \int _{\mathbb{R}^{n}}\rho _{m}(x) \bigl\vert u_{t}(t,x) \bigr\vert ^{2}\,dx \\ &\qquad {}+\varepsilon \biggl(1- \frac{1}{4}\gamma \biggr) \int _{\mathbb{R}^{n}}\rho _{m}(x) \bigl\vert \Delta u(t,x) \bigr\vert ^{2}\,dx \\ &\qquad {}+\frac{1}{2}\varepsilon \biggl(\nu -\frac{1}{2}\nu \gamma - \varepsilon \alpha +\frac{1}{4}\varepsilon ^{2}\gamma \biggr) \int _{\mathbb{R}^{n}} \rho _{m}(x) \bigl\vert u(t,x) \bigr\vert ^{2}\,dx \\ &\qquad {}+\frac{1}{2}\varepsilon \gamma \int _{ \mathbb{R}^{n}}\rho _{m}(x)F\bigl(x,u(t,x)\bigr)\,dx \\ &\quad \leq c_{14} \int _{\mathbb{R}^{n}}\rho _{m}(x) \bigl( \bigl\vert g(t,x) \bigr\vert ^{2}+ \bigl\vert \varphi _{1}(x) \bigr\vert + \bigl\vert \varphi _{3}(x) \bigr\vert + \bigl\vert \zeta _{\delta}(\theta _{t} \omega )\varphi _{6}(x) \bigr\vert ^{2} \\ &\qquad {} + \bigl\vert \zeta _{\delta}(\theta _{t} \omega ) \varphi _{5}(x) \bigr\vert ^{2+\frac{4}{p}} \bigr)\,dx \\ &\qquad {}+\frac{c_{14}}{m}\bigl( \Vert u \Vert + \Vert \nabla u \Vert + \Vert u_{t} \Vert + \Vert \nabla u_{t} \Vert \bigr) \Vert \Delta u \Vert . \end{aligned}$$

(4.35)

By (4.7) and (4.35) we have

$$\begin{aligned} &\frac{d}{dt} \int _{\mathbb{R}^{n}}\rho _{m}(x) \bigl( \bigl\vert u_{t}(t,x) \bigr\vert ^{2}+ \nu \bigl\vert u(t,x) \bigr\vert ^{2}+ \bigl\vert \Delta u(t,x) \bigr\vert ^{2}+2F \bigl(x,u(t,x)\bigr)+\varepsilon u(t,x)u_{t}(t,x) \bigr)\,dx \\ &\qquad {}+\frac{1}{4}\varepsilon \gamma \int _{\mathbb{R}^{n}}\rho _{m}(x) \bigl( \bigl\vert u_{t}(t,x) \bigr\vert ^{2}+\nu \bigl\vert u(t,x) \bigr\vert ^{2}+ \bigl\vert \Delta u(t,x) \bigr\vert ^{2} \\ &\qquad {}+2F \bigl(x,u(t,x)\bigr)+ \varepsilon u(t,x)u_{t}(t,x) \bigr)\,dx \\ &\quad \leq c_{14} \int _{\mathbb{R}^{n}}\rho _{m}(x) \bigl( \bigl\vert g(t,x) \bigr\vert ^{2}+ \bigl\vert \varphi _{1}(x) \bigr\vert + \bigl\vert \varphi _{3}(x) \bigr\vert + \bigl\vert \zeta _{\delta}(\theta _{t} \omega )\varphi _{6}(x) \bigr\vert ^{2} \\ &\qquad {} + \bigl\vert \zeta _{\delta}(\theta _{t} \omega ) \varphi _{5}(x) \bigr\vert ^{2+\frac{4}{p}} \bigr)\,dx \\ &\qquad {}+\frac{c_{14}}{m}\bigl( \Vert u \Vert + \Vert \nabla u \Vert + \Vert u_{t} \Vert + \Vert \nabla u_{t} \Vert \bigr) \Vert \Delta u \Vert . \end{aligned}$$

(4.36)

Multiplying (4.36) by $e^{\frac{1}{4}\varepsilon \gamma t}$, and then integrating the inequality $[\tau -t,\tau ]$, after replacing ω by $\theta _{-\tau}\omega$, we obtain

$$\begin{aligned} & \int _{\mathbb{R}^{n}}\rho _{m}(x) \bigl( \bigl\vert u_{t}(\tau ,\tau -t,\theta _{- \tau}\omega ,u_{1,0}) \bigr\vert ^{2}+\nu \bigl\vert u(\tau ,\tau -t,\theta _{-\tau} \omega ,u_{0}) \bigr\vert ^{2} \\ &\qquad {}+ \bigl\vert \Delta u(\tau ,\tau -t,\theta _{-\tau}\omega ,u_{0}) \bigr\vert ^{2} \\ &\qquad {}+2F\bigl(x,u(\tau ,\tau -t,\theta _{-\tau}\omega ,u_{0}) \bigr)+\varepsilon u( \tau ,\tau -t,\theta _{-\tau}\omega ,u_{0})u_{t}(\tau ,\tau -t, \theta _{-\tau}\omega ,u_{1,0}) \bigr)\,dx \\ &\quad \leq e^{-\frac{1}{4}\varepsilon \gamma t} \int _{\mathbb{R}^{n}}\rho _{m}(x) \bigl( \vert u_{1,0} \vert ^{2}+ \nu \vert u_{0} \vert ^{2}+ \vert \Delta u_{0} \vert ^{2}+2F \bigl(x,u_{0}(x)\bigr)+\varepsilon u_{0}(x)u_{1,0}(x) \bigr)\,dx \\ &\qquad {}+c_{14} \int ^{\tau}_{\tau -t}e^{\frac{1}{4}\varepsilon \gamma (s- \tau )} \int _{\mathbb{R}^{n}}\rho _{m}(x) \bigl( \bigl\vert g(s,x) \bigr\vert ^{2}+ \bigl\vert \varphi _{1}(x) \bigr\vert + \bigl\vert \varphi _{3}(x) \bigr\vert \bigr)\,dx\,ds \\ &\qquad {}+c_{14} \int ^{\tau}_{\tau -t}e^{\frac{1}{4}\varepsilon \gamma (s- \tau )} \int _{\mathbb{R}^{n}}\rho _{m}(x) \bigl( \bigl\vert \zeta _{\delta}(\theta _{s- \tau}\omega )\varphi _{6}(x) \bigr\vert ^{2} + \bigl\vert \zeta _{\delta}(\theta _{s-\tau} \omega )\varphi _{5}(x) \bigr\vert ^{2+\frac{4}{p}} \bigr)\,dx\,ds \\ &\qquad {}+\frac{2c_{14}}{m} \int ^{\tau}_{\tau -t}e^{\frac{1}{4}\varepsilon \gamma (s-\tau )}\bigl( \bigl\Vert u(\tau ,\tau -t,\theta _{-\tau}\omega ,u_{0}) \bigr\Vert ^{2}_{H^{2}( \mathbb{R}^{n})} \\ &\qquad {}+ \bigl\Vert u_{t}(\tau ,\tau -t, \theta _{-\tau}\omega ,u_{1,0}) \bigr\Vert ^{2}_{H^{1}(\mathbb{R}^{n})} \bigr)\,ds. \end{aligned}$$

(4.37)

Next, we estimate the right-hand side of (4.37). By (4.16), we know that there exists $T_{1}(\eta ,\tau ,\omega ,D)>0$ such that for all $t\geq T_{1}$,

$$\begin{aligned} &e^{-\frac{1}{4}\varepsilon \gamma t} \int _{\mathbb{R}^{n}}\rho _{m}(x) \bigl( \vert u_{1,0} \vert ^{2}+ \nu \vert u_{0} \vert ^{2}+ \vert \Delta u_{0} \vert ^{2} \\ &\quad {}+2F \bigl(x,u_{0}(x)\bigr)+\varepsilon u_{0}(x)u_{1,0}(x) \bigr)\,dx< \eta . \end{aligned}$$

(4.38)

For the second and the third terms on the right-hand side of (4.37) we obtain

$$\begin{aligned} &c_{14} \int ^{\tau}_{\tau -t}e^{\frac{1}{4}\varepsilon \gamma (s- \tau )} \int _{\mathbb{R}^{n}}\rho _{m}(x) \bigl( \bigl\vert g(s,x) \bigr\vert ^{2}+ \bigl\vert \varphi _{1}(x) \bigr\vert + \bigl\vert \varphi _{3}(x) \bigr\vert \bigr)\,dx\,ds \\ &\qquad {}+c_{14} \int ^{\tau}_{\tau -t}e^{\frac{1}{4}\varepsilon \gamma (s- \tau )} \int _{\mathbb{R}^{n}}\bigl(\rho _{m}(x) \bigl\vert \zeta _{\delta}(\theta _{s- \tau}\omega )\varphi _{6}(x) \bigr\vert ^{2} + \bigl\vert \zeta _{\delta}(\theta _{s-\tau} \omega )\varphi _{5}(x) \bigr\vert ^{2+\frac{4}{p}} \bigr)\,dx\,ds \\ &\quad \leq c_{14} \int ^{\tau}_{-\infty}e^{\frac{1}{4}\varepsilon \gamma (s- \tau )} \int _{ \vert x \vert \geq \frac{1}{2}m}\bigl( \bigl\vert g(s,x) \bigr\vert ^{2}+ \bigl\vert \varphi _{1}(x) \bigr\vert + \bigl\vert \varphi _{3}(x) \bigr\vert \bigr)\,dx\,ds \\ &\qquad {}+c_{14} \int ^{\tau}_{-\infty}e^{\frac{1}{4}\varepsilon \gamma (s- \tau )} \int _{ \vert x \vert \geq \frac{1}{2}m}\bigl( \bigl\vert \zeta _{\delta}(\theta _{s-\tau} \omega )\varphi _{6}(x) \bigr\vert ^{2} + \bigl\vert \zeta _{\delta}(\theta _{s-\tau} \omega )\varphi _{5}(x) \bigr\vert ^{2+\frac{4}{p}}\bigr)\,dx\,ds \\ &\quad \leq c_{14} \int ^{\tau}_{-\infty}e^{\frac{1}{4}\varepsilon \gamma (s- \tau )} \int _{ \vert x \vert \geq \frac{1}{2}m}\bigl( \bigl\vert g(s,x) \bigr\vert ^{2}+ \bigl\vert \varphi _{1}(x) \bigr\vert + \bigl\vert \varphi _{3}(x) \bigr\vert \bigr)\,dx\,ds \\ &\qquad {}+c_{14} \int ^{0}_{-\infty}e^{\frac{1}{4}\varepsilon \gamma s } \bigl\vert \zeta _{\delta}(\theta _{s}\omega ) \bigr\vert ^{2}\,ds \int _{ \vert x \vert \geq \frac{1}{2}m} \bigl\vert \varphi _{6}(x) \bigr\vert ^{2}\,dx \\ &\qquad {}+c_{14} \int ^{0}_{-\infty}e^{\frac{1}{4}\varepsilon \gamma s } \bigl\vert \zeta _{\delta}(\theta _{s}\omega ) \bigr\vert ^{2+\frac{4}{p}}\,ds \int _{ \vert x \vert \geq \frac{1}{2}m} \bigl\vert \varphi _{5}(x) \bigr\vert ^{2+\frac{4}{p}}\,dx. \end{aligned}$$

(4.39)

By (4.8) and with the conditions of $\varphi _{i}(x)$ ($i=1,3,5,6$) satisfied, we know that there exists $m_{1}=m_{1}(\eta ,\tau ,\omega )\geq 1$ such that for all $m\geq m_{1}$, the right-hand of side of (4.39) is bounded by η, i.e.,

$$\begin{aligned} &c_{14} \int ^{\tau}_{\tau -t}e^{\frac{1}{4}\varepsilon \gamma (s- \tau )} \int _{\mathbb{R}^{n}}\rho _{m}(x) \bigl( \bigl\vert g(s,x) \bigr\vert ^{2}+ \bigl\vert \varphi _{1}(x) \bigr\vert + \bigl\vert \varphi _{3}(x) \bigr\vert \bigr)\,dx\,ds \\ &\qquad {}+c_{14} \int ^{\tau}_{\tau -t}e^{\frac{1}{4}\varepsilon \gamma (s- \tau )} \int _{\mathbb{R}^{n}}\bigl(\rho _{m}(x) \bigl\vert \zeta _{\delta}(\theta _{s- \tau}\omega )\varphi _{6}(x) \bigr\vert ^{2} + \bigl\vert \zeta _{\delta}(\theta _{s-\tau} \omega )\varphi _{5}(x) \bigr\vert ^{2+\frac{4}{p}} \bigr)\,dx\,ds \\ &\quad < \eta . \end{aligned}$$

(4.40)

For the last term in (4.37), by Lemma 4.1 and Lemma 4.3, we know that there exists $T_{2}(\eta ,\tau ,\omega ,D)\geq T_{1}$ such that for all $t\geq T_{2}$,

$$\begin{aligned} &\frac{2c_{14}}{m} \int ^{\tau}_{\tau -t}e^{\frac{1}{4}\varepsilon \gamma (s-\tau )}\bigl( \bigl\Vert u(\tau ,\tau -t,\theta _{-\tau}\omega ,u_{0}) \bigr\Vert ^{2}_{H^{2}( \mathbb{R}^{n})} \\ &\quad {}+ \bigl\Vert u_{t}(\tau ,\tau -t, \theta _{-\tau}\omega ,u_{1,0}) \bigr\Vert ^{2}_{H^{1}(\mathbb{R}^{n})} \bigr)\,ds\leq \frac{c_{15}}{m}, \end{aligned}$$

where $c_{15}>0$ depends only on α, ν, γ, ε, τ, and ω, but not on m. Thus, there exists $m_{2}=m_{2}(\eta ,\tau ,\omega )\geq m_{1}$ such that for all $m\geq m_{2}$ and $t\geq T_{2}$,

$$\begin{aligned} &\frac{2c_{14}}{m} \int ^{\tau}_{\tau -t}e^{\frac{1}{4}\varepsilon \gamma (s-\tau )}\bigl( \bigl\Vert u(\tau ,\tau -t,\theta _{-\tau}\omega ,u_{0}) \bigr\Vert ^{2}_{H^{2}( \mathbb{R}^{n})} \\ &\quad {} + \bigl\Vert u_{t}(\tau ,\tau -t, \theta _{-\tau}\omega ,u_{1,0}) \bigr\Vert ^{2}_{H^{1}(\mathbb{R}^{n})} \bigr)\,ds\leq \eta . \end{aligned}$$

(4.41)

By (4.37), (4.38), (4.40), and (4.41) we see that for all $m\geq m_{2}$ and $t\geq T_{2}$,

$$\begin{aligned} & \int _{\mathbb{R}^{n}}\rho _{m}(x) \bigl( \bigl\vert u_{t}(\tau ,\tau -t,\theta _{- \tau}\omega ,u_{1,0}) \bigr\vert ^{2}+\nu \bigl\vert u(\tau ,\tau -t,\theta _{-\tau} \omega ,u_{0}) \bigr\vert ^{2} \\ &\qquad {}+ \bigl\vert \Delta u(\tau ,\tau -t,\theta _{-\tau}\omega ,u_{0}) \bigr\vert ^{2} \\ &\qquad {}+2F\bigl(x,u(\tau ,\tau -t,\theta _{-\tau}\omega ,u_{0}) \bigr)+\varepsilon u( \tau ,\tau -t,\theta _{-\tau}\omega ,u_{0})u_{t}(\tau ,\tau -t, \theta _{-\tau}\omega ,u_{1,0}) \bigr)\,dx \\ &\quad < 3\eta . \end{aligned}$$

(4.42)

By (4.7) we have

$$\begin{aligned} &\varepsilon \int _{\mathbb{R}^{n}}\rho _{m}(x)u(\tau ,\tau -t, \theta _{-\tau}\omega ,u_{0})u_{t}(\tau ,\tau -t,\theta _{-\tau} \omega ,u_{1,0})\,dx \\ &\quad \leq \frac{1}{2}\nu \int _{\mathbb{R}^{n}}\rho _{m}(x) \bigl\vert u(\tau ,\tau -t, \theta _{-\tau}\omega ,u_{0}) \bigr\vert ^{2}\,dx \\ &\qquad {}+ \frac{1}{2} \int _{\mathbb{R}^{n}} \rho _{m}(x) \bigl\vert u_{t}(\tau ,\tau -t,\theta _{-\tau}\omega ,u_{1,0}) \bigr\vert ^{2}\,dx, \end{aligned}$$

which together with (4.2) and (4.42) yields that for all $m\geq m_{2}$ and $t\geq T_{2}$,

$$\begin{aligned} & \int _{\mathbb{R}^{n}}\rho _{m}(x) (\frac{1}{2} \bigl\vert u_{t}(\tau , \tau -t,\theta _{-\tau}\omega ,u_{1,0}) \bigr\vert ^{2}+\frac{1}{2}\nu \bigl\vert u(\tau , \tau -t,\theta _{-\tau}\omega ,u_{0}) \bigr\vert ^{2} \\ &\qquad {}+ \bigl\vert \Delta u(\tau ,\tau -t, \theta _{-\tau} \omega ,u_{0}) \bigr\vert ^{2}\,dx \\ &\quad \leq 3\eta +2 \int _{\mathbb{R}^{n}}\rho _{m}(x)\varphi _{1}(x)\,dx. \end{aligned}$$

(4.43)

Since $\varphi _{1}\in L^{1}(\mathbb{R}^{n})$, there exists $m_{3}=m_{3}(\eta ,\tau ,\omega )\geq m_{2}$ such that for all $m\geq m_{3}$,

$$ 2 \int _{\mathbb{R}^{n}}\rho _{m}(x)\varphi _{1}(x)\,dx=2 \int _{ \vert x \vert \geq \frac{1}{2}m}\rho _{m}(x)\varphi _{1}(x)\,dx \leq 2 \int _{ \vert x \vert \geq \frac{1}{2}m} \bigl\vert \varphi _{1}(x) \bigr\vert \,dx< \eta . $$

(4.44)

From (4.43) and (4.44) we obtain, for all $m\geq m_{3}$ and $t\geq T_{2}$,

$$\begin{aligned} & \int _{ \vert x \vert \geq m}\rho _{m}(x) (\frac{1}{2} \bigl\vert u_{t}(\tau ,\tau -t, \theta _{-\tau}\omega ,u_{1,0}) \bigr\vert ^{2}+\frac{1}{2}\nu \bigl\vert u(\tau ,\tau -t, \theta _{-\tau}\omega ,u_{0}) \bigr\vert ^{2} \\ &\qquad {}+ \bigl\vert \Delta u(\tau ,\tau -t,\theta _{- \tau}\omega ,u_{0}) \bigr\vert ^{2}\,dx \\ &\quad \leq \int _{\mathbb{R}^{n}}\rho _{m}(x) (\frac{1}{2} \bigl\vert u_{t}(\tau , \tau -t,\theta _{-\tau}\omega ,u_{1,0}) \bigr\vert ^{2}+\frac{1}{2}\nu \bigl\vert u(\tau , \tau -t,\theta _{-\tau}\omega ,u_{0}) \bigr\vert ^{2} \\ &\qquad {}+ \bigl\vert \Delta u(\tau ,\tau -t, \theta _{-\tau} \omega ,u_{0}) \bigr\vert ^{2}\,dx \\ &\quad < 4\eta . \end{aligned}$$

 □

5 Existence of random attractors

In this section, we present the existence and uniqueness of $\mathcal{D}$-pullback random attractors of (1.1).

Let u be the solution of (1.1). Denote $u=\tilde{v}+v$, where ṽ and v are the solutions of the following equations, respectively,

$$ \textstyle\begin{cases} \tilde{v}_{tt}+\alpha \tilde{v}_{t}+\Delta ^{2}\tilde{v}+\nu \tilde{v}=g(t), \quad t>\tau , \\ \tilde{v}(\tau )=u_{0}, \qquad \tilde{v}_{t}(\tau )=u_{1,0}, \end{cases} $$

(5.1)

and

$$ \textstyle\begin{cases} v_{tt}+\alpha v_{t}+\Delta ^{2}v+\nu v =-f(x,u)+h(t,x,u)\zeta _{ \delta}(\theta _{t}\omega ), \quad t>\tau , \\ v(\tau )=0,\qquad v_{t}(\tau )=0. \end{cases} $$

(5.2)

Lemma 5.1

Suppose (4.7) and (4.8) hold. Then, for every $\tau \in \mathbb{R}$, $\omega \in \Omega$ and $D\in \mathcal{D}$, there exists $T=T(\tau ,\omega ,D)>0$ such that for all $t\geq T$ and $r\in [-t,0]$, the solution ṽ of (5.1) satisfies

$$\begin{aligned} & \bigl\Vert \tilde{v}(\tau +r,\tau -t,\theta _{-\tau}\omega ,u_{0}) \bigr\Vert ^{2}_{H^{2}( \mathbb{R}^{n})}+ \bigl\Vert \tilde{v}_{r}(\tau +r,\tau -t,\theta _{-\tau} \omega ,u_{1,0}) \bigr\Vert ^{2} \\ &\quad \leq e^{-\frac{1}{2}\varepsilon r}M_{2} \biggl(1+ \int ^{0}_{\infty }e^{ \frac{1}{2}\varepsilon s} \bigl\Vert g(s+ \tau ) \bigr\Vert ^{2}\,ds \biggr), \end{aligned}$$

where $(u_{0},u_{1,0})\in D(\tau -t,\theta _{-t}\omega )$ and $M_{2}$ is a positive number independent of τ, ω, and D.

Proof

From (3.8), (3.9), and (5.1) we see that

$$\begin{aligned} &\frac{d}{dt}\bigl( \Vert \tilde{v}_{t} \Vert ^{2}+ \Vert \Delta \tilde{v} \Vert ^{2}+\nu \Vert \tilde{v} \Vert ^{2}+\varepsilon \bigl(\tilde{v}(t), \tilde{v}_{t}(t)\bigr)\bigr)+(2 \alpha -\varepsilon ) \Vert \tilde{v}_{t} \Vert ^{2} \\ &\qquad {}+\varepsilon \Vert \Delta \tilde{v} \Vert ^{2}+ \varepsilon \nu \Vert \tilde{v} \Vert ^{2}+ \varepsilon \alpha \bigl(\tilde{v}(t), \tilde{v}_{t}(t)\bigr) \\ &\quad =\bigl(g(t),\varepsilon \tilde{v}(t)+2\tilde{v}_{t}(t)\bigr) \\ &\quad \leq \varepsilon \bigl\Vert g(t) \bigr\Vert \bigl\Vert \tilde{v}(t) \bigr\Vert +2 \bigl\Vert g(t) \bigr\Vert \bigl\Vert \tilde{v}_{t}(t) \bigr\Vert \\ &\quad \leq \frac{1}{2}\varepsilon ^{2} \bigl\Vert \tilde{v}(t) \bigr\Vert ^{2}+\alpha \bigl\Vert \tilde{v}_{t}(t) \bigr\Vert ^{2}+\biggl(\frac{1}{2}+\alpha ^{-1}\biggr) \bigl\Vert g(t) \bigr\Vert ^{2}. \end{aligned}$$

(5.3)

In addition, we obtain

$$ \biggl\vert \biggl(\alpha -\frac{1}{2}\varepsilon \biggr)\varepsilon \bigl(\tilde{v}(t),\tilde{v}_{t}(t)\bigr) \biggr\vert \leq \frac{1}{2}\biggl(\alpha -\frac{1}{2}\varepsilon \biggr) \bigl( \varepsilon ^{2} \bigl\Vert \tilde{v}(t) \bigr\Vert ^{2}+ \bigl\Vert \tilde{v}_{t}(t) \bigr\Vert ^{2}\bigr). $$

(5.4)

By (5.3) and (5.4) we have

$$\begin{aligned} &\frac{d}{dt}\bigl( \Vert \tilde{v}_{t} \Vert ^{2}+ \Vert \Delta \tilde{v} \Vert ^{2}+\nu \Vert \tilde{v} \Vert ^{2}+\varepsilon \bigl(\tilde{v}(t), \tilde{v}_{t}(t)\bigr)\bigr)+\biggl( \frac{1}{2}\alpha - \frac{3}{4}\varepsilon \biggr) \Vert \tilde{v}_{t} \Vert ^{2} \\ &\qquad {}+\varepsilon \Vert \Delta \tilde{v} \Vert ^{2}+ \varepsilon \biggl(\nu -\frac{1}{2} \varepsilon -\frac{1}{2}\varepsilon \alpha + \frac{1}{4} \varepsilon ^{2}\biggr) \Vert \tilde{v} \Vert ^{2}+\frac{1}{2}\varepsilon ^{2}\bigl(\tilde{v}(t), \tilde{v}_{t}(t)\bigr) \\ &\quad \leq \biggl(\frac{1}{2}+\alpha ^{-1}\biggr) \bigl\Vert g(t) \bigr\Vert ^{2}, \end{aligned}$$

which can be rewritten as

$$\begin{aligned} &\frac{d}{dt}\bigl( \Vert \tilde{v}_{t} \Vert ^{2}+ \Vert \Delta \tilde{v} \Vert ^{2}+\nu \Vert \tilde{v} \Vert ^{2}+\varepsilon \bigl(\tilde{v}(t), \tilde{v}_{t}(t)\bigr)\bigr) \\ &\qquad {}+\frac{1}{2}\varepsilon \bigl( \Vert \tilde{v}_{t} \Vert ^{2}+ \Vert \Delta \tilde{v} \Vert ^{2}+ \nu \Vert \tilde{v} \Vert ^{2}+\varepsilon \bigl(\tilde{v}(t), \tilde{v}_{t}(t)\bigr)\bigr) \\ &\qquad {}+\biggl(\frac{1}{2}\alpha -\frac{5}{4}\varepsilon \biggr) \Vert \tilde{v}_{t} \Vert ^{2}+ \frac{1}{2}\varepsilon \Vert \Delta \tilde{v} \Vert ^{2}+\frac{1}{2} \varepsilon \biggl(\nu - \varepsilon - \varepsilon \alpha +\frac{1}{2} \varepsilon ^{2}\biggr) \Vert \tilde{v} \Vert ^{2} \\ &\quad \leq \biggl(\frac{1}{2}+\alpha ^{-1}\biggr) \bigl\Vert g(t) \bigr\Vert ^{2}. \end{aligned}$$

(5.5)

It follows from (4.7) and (5.5) that

$$\begin{aligned} &\frac{d}{dt}\bigl( \Vert \tilde{v}_{t} \Vert ^{2}+ \Vert \Delta \tilde{v} \Vert ^{2}+\nu \Vert \tilde{v} \Vert ^{2}+\varepsilon \bigl(\tilde{v}(t), \tilde{v}_{t}(t)\bigr)\bigr) \\ &\qquad {}+\frac{1}{2}\varepsilon \bigl( \Vert \tilde{v}_{t} \Vert ^{2}+ \Vert \Delta \tilde{v} \Vert ^{2}+ \nu \Vert \tilde{v} \Vert ^{2}+\varepsilon \bigl(\tilde{v}(t), \tilde{v}_{t}(t)\bigr)\bigr) \\ &\quad \leq \biggl(\frac{1}{2}+\alpha ^{-1}\biggr) \bigl\Vert g(t) \bigr\Vert ^{2}. \end{aligned}$$

(5.6)

Applying Gronwall’s lemma to (5.6), we obtain for all $\tau \in \mathbb{R}$, $t\geq 0$, $r\in [-t,0]$ and $\omega \in \Omega$,

$$\begin{aligned} & \bigl\Vert \tilde{v}_{r}(\tau +r,\tau -t,\theta _{-\tau} \omega ,u_{1,0}) \bigr\Vert ^{2}+ \bigl\Vert \Delta \tilde{v}(\tau +r,\tau -t,\theta _{-\tau}\omega ,u_{0}) \bigr\Vert ^{2} \\ &\qquad {}+ \nu \bigl\Vert \tilde{v}(\tau +r,\tau -t,\theta _{-\tau}\omega ,u_{0}) \bigr\Vert ^{2} \\ &\qquad {}+\varepsilon \bigl(\tilde{v}(\tau +r,\tau -t,\theta _{-\tau}\omega ,u_{0}), \tilde{v}_{r}(\tau +r,\tau -t,\theta _{-\tau}\omega ,u_{1,0})\bigr) \\ &\quad \leq e^{-\frac{1}{2}\varepsilon r}e^{-\frac{1}{2}\varepsilon t} \bigl( \Vert u_{1,0} \Vert ^{2}+\nu \Vert u_{0} \Vert ^{2}+ \Vert \Delta u_{0} \Vert ^{2}+ \varepsilon (u_{0},u_{1,0}) \bigr) \\ &\qquad {}+\biggl(\frac{1}{2}+\alpha ^{-1}\biggr)e^{-\frac{1}{2}\varepsilon r} \int ^{\tau +r}_{ \tau -t}e^{\frac{1}{2}\varepsilon (s-\tau )} \bigl\Vert g(s) \bigr\Vert ^{2}\,ds. \end{aligned}$$

(5.7)

By (4.7) we have

$$\begin{aligned} &\varepsilon \bigl(\tilde{v}(\tau +r,\tau -t,\theta _{-\tau}\omega ,u_{0}), \tilde{v}_{r}(\tau +r,\tau -t,\theta _{-\tau}\omega ,u_{1,0})\bigr) \\ &\quad \leq \frac{1}{2}\varepsilon \bigl\Vert \tilde{v}(\tau +r,\tau -t, \theta _{- \tau}\omega ,u_{0}) \bigr\Vert ^{2}+ \frac{1}{2}\varepsilon \bigl\Vert \tilde{v}_{r}( \tau +r,\tau -t,\theta _{-\tau}\omega ,u_{1,0}) \bigr\Vert ^{2} \\ &\quad \leq \frac{1}{2}\nu \bigl\Vert \tilde{v}(\tau +r,\tau -t,\theta _{-\tau} \omega ,u_{0}) \bigr\Vert ^{2}+ \frac{1}{2} \bigl\Vert \tilde{v}_{r}(\tau +r,\tau -t, \theta _{-\tau}\omega ,u_{1,0}) \bigr\Vert ^{2}. \end{aligned}$$

(5.8)

By (5.7) and (5.8) we see that for all $\tau \in \mathbb{R}$, $t\geq 0$, $r\in [-t,0]$ and $\omega \in \Omega$,

$$\begin{aligned} &\frac{1}{2} \bigl\Vert \tilde{v}_{r}(\tau +r,\tau -t,\theta _{-\tau}\omega ,u_{1,0}) \bigr\Vert ^{2}+ \bigl\Vert \Delta \tilde{v}(\tau +r,\tau -t,\theta _{-\tau}\omega ,u_{0}) \bigr\Vert ^{2} \\ &\qquad {}+\frac{1}{2}\nu \bigl\Vert \tilde{v}(\tau +r,\tau -t,\theta _{-\tau} \omega ,u_{0}) \bigr\Vert ^{2} \\ &\quad \leq e^{-\frac{1}{2}\varepsilon r}e^{-\frac{1}{2}\varepsilon t} \bigl( \Vert u_{1,0} \Vert ^{2}+\nu \Vert u_{0} \Vert ^{2}+ \Vert \Delta u_{0} \Vert ^{2}+ \varepsilon (u_{0},u_{1,0}) \bigr) \\ &\qquad {}+\biggl(\frac{1}{2}+\alpha ^{-1}\biggr)e^{-\frac{1}{2}\varepsilon r} \int ^{\tau +r}_{ \tau -t}e^{\frac{1}{2}\varepsilon (s-\tau )} \bigl\Vert g(s) \bigr\Vert ^{2}\,ds. \end{aligned}$$

(5.9)

Similar to (4.16), one can verify that

$$ e^{-\frac{1}{2}\varepsilon t} \bigl( \Vert u_{1,0} \Vert ^{2}+\nu \Vert u_{0} \Vert ^{2}+ \Vert \Delta u_{0} \Vert ^{2}+ \varepsilon (u_{0},u_{1,0}) \bigr) \rightarrow 0,\quad \text{as } t\rightarrow \infty , $$

which along with (5.9) yields the desired result. □

Based on Lemma 5.1, we infer that system (5.1) has a tempered pullback random absorbing set.

Lemma 5.2

Suppose (4.8) and (4.9) hold, then (5.1) possesses a closed measurable $\mathcal{D}$-pullback absorbing set $B_{1}=\{B_{1}(\tau ,\omega ):\tau \in \mathbb{R},\omega \in \Omega \}\in \mathcal{D}$, which is given by

$$ B_{1}(\tau ,\omega )=\bigl\{ (u_{0},u_{1,0})\in H^{2}\bigl(\mathbb{R}^{n}\bigr) \times L^{2}\bigl( \mathbb{R}^{n}\bigr): \Vert u_{0} \Vert ^{2}_{H^{2}(\mathbb{R}^{n})}+ \Vert u_{1,0} \Vert ^{2} \leq L_{1}(\tau ,\omega )\bigr\} , $$

(5.10)

where

$$ L_{1}(\tau ,\omega )=M_{2}+M_{2} \int ^{0}_{-\infty}e^{\frac{1}{2} \varepsilon s} \bigl\Vert g(s+ \tau ) \bigr\Vert ^{2}\,ds. $$

Lemma 5.3

Suppose (4.8) and (4.9) hold, then the sequence of the solutions to (5.1)

$$ \bigl\{ \tilde{v}\bigl(\tau ,\tau -t_{n},\theta _{-\tau}\omega ,u^{(n)}_{0}\bigr), \tilde{v}_{t}\bigl(\tau ,\tau -t_{n},\theta _{-\tau}\omega ,u^{(n)}_{1,0} \bigr) \bigr\} ^{\infty}_{n=1} $$

converges in $H^{2}(\mathbb{R}^{n})\times L^{2}(\mathbb{R}^{n})$ for any $\tau \in \mathbb{R}$, $\omega \in \Omega$, $D\in \mathcal{D}$, $t_{n} \rightarrow \infty$ monotonically, and $(u^{(n)}_{0},u^{(n)}_{1,0})\in D(\tau -t_{n},\theta _{-t_{n}}\omega )$.

Proof

Let $m>n$ and

$$\begin{aligned} &v_{n,m}(t,\tau -t_{n},\theta _{-\tau}\omega ) \\ &\quad =\tilde{v}\bigl(t,\tau -t_{n},\theta _{-\tau}\omega ,u^{(n)}_{0}\bigr)- \tilde{v}\bigl(t,\tau -t_{m}, \theta _{-\tau}\omega ,u^{(m)}_{0}\bigr) \\ &\quad =\tilde{v}\bigl(t,\tau -t_{n},\theta _{-\tau}\omega ,u^{(n)}_{0}\bigr)- \tilde{v}(t,\tau -t_{n},\theta _{-\tau}\omega ,\tilde{v}\bigl(\tau -t_{n}, \tau -t_{m},\theta _{-\tau}\omega ,u^{(m)}_{0} \bigr) \end{aligned}$$

(5.11)

for $t\geq \tau -t_{n}$.

By (5.1) we obtain

$$ \textstyle\begin{cases} \partial ^{2}_{tt}v_{n,m}(t)+\alpha \partial _{t}v_{n,m}(t)+\Delta ^{2}v_{n,m}(t)+ \nu v_{n,m}(t)=0, \quad t>\tau -t_{n}, \\ v_{n,m}(\tau -t_{n})=u^{(n)}_{0}-\tilde{v}(\tau -t_{n},\tau -t_{m}, \theta _{-\tau}\omega ,u^{(m)}_{0}), \\ \partial _{t}v_{n,m}(\tau -t_{n})=u^{(n)}_{1,0}- \tilde{v}_{t}. \end{cases} $$

(5.12)

Similar to (5.9) with $r = 0$, $t = t_{n}$, and $g = 0$, we obtain

$$\begin{aligned} &\frac{1}{2} \bigl\Vert \partial _{t}v_{n,m}(\tau ,\tau -t_{n},\theta _{-\tau} \omega ) \bigr\Vert ^{2}+ \bigl\Vert \Delta v_{n,m}(\tau ,\tau -t_{n},\theta _{-\tau} \omega ) \bigr\Vert ^{2}+ \frac{1}{2}\nu v_{n,m}(\tau ,\tau -t_{n},\theta _{- \tau}\omega ) \Vert ^{2} \\ &\quad \leq e^{-\frac{1}{2}\varepsilon t_{n}}\bigl( \bigl\Vert \partial _{t}v_{n,m}( \tau -t_{n}) \bigr\Vert ^{2}+ \bigl\Vert v_{n,m}(\tau -t_{n}) \bigr\Vert ^{2}+ \bigl\Vert \Delta v_{n,m}(\tau -t_{n}) \bigr\Vert ^{2}\bigr), \end{aligned}$$

(5.13)

which together with (5.12)₂, gives

$$\begin{aligned} & \bigl\Vert \partial _{t}v_{n,m}(\tau ,\tau -t_{n},\theta _{-\tau}\omega ) \bigr\Vert ^{2}+2 \bigl\Vert \Delta v_{n,m}(\tau ,\tau -t_{n},\theta _{-\tau}\omega ) \bigr\Vert ^{2}+ \nu v_{n,m}(\tau ,\tau -t_{n},\theta _{-\tau}\omega ) \Vert ^{2} \\ &\quad \leq 2e^{-\frac{1}{2}\varepsilon t_{n}}(\|\tilde{v}_{t}(\tau -t_{n}, \tau -t_{m},\theta _{-\tau}\omega ,u^{(m)}_{1,0} \Vert ^{2}+ \|\tilde{v}\bigl( \tau -t_{n},\tau -t_{m},\theta _{-\tau}\omega ,u^{(m)}_{0} \|^{2}_{H^{2}}\bigr) \\ &\qquad {}+2e^{-\frac{1}{2}\varepsilon t_{n}}\bigl( \bigl\Vert u^{(n)}_{1,0} \bigr\Vert ^{2}+ \bigl\Vert u^{(n)}_{0} \bigr\Vert ^{2}+ \bigl\Vert \Delta u^{(n)}_{0} \bigr\Vert ^{2}\bigr). \end{aligned}$$

(5.14)

By (5.9) with $r = -t_{n}$, and $t = t_{m}$, we obtain

$$\begin{aligned} & \bigl\Vert \tilde{v}_{t}\bigl(\tau -t_{n},\tau -t_{m},\theta _{-\tau}\omega ,u^{(m)}_{1,0} \bigr) \bigr\Vert ^{2}+2 \bigl\Vert \Delta \tilde{v}\bigl(\tau -t_{n},\tau -t_{m},\theta _{-\tau} \omega ,u^{(m)}_{0}\bigr) \bigr\Vert ^{2} \\ &\qquad {}+\nu \bigl\Vert \tilde{v}\bigl(\tau -t_{n},\tau -t_{m}, \theta _{-\tau}\omega ,u^{(m)}_{0}\bigr) \bigr\Vert ^{2} \\ &\quad \leq 2e^{\frac{1}{2}\varepsilon t_{n}}e^{-\frac{1}{2}\varepsilon t_{m}} \bigl( \bigl\Vert u^{(n)}_{1,0} \bigr\Vert ^{2}+\nu \bigl\Vert u^{(n)}_{0} \bigr\Vert ^{2}+ \bigl\Vert \Delta u^{(n)}_{0} \bigr\Vert ^{2}+ \varepsilon \bigl(u^{(n)}_{0},u^{(n)}_{1,0} \bigr) \bigr) \\ &\qquad {}+\bigl(1+2\alpha ^{-1}\bigr)e^{\frac{1}{2}\varepsilon t_{n}} \int ^{\tau -t_{n}}_{ \tau -t_{m}}e^{\frac{1}{2}\varepsilon (s-\tau )} \bigl\Vert g(s) \bigr\Vert ^{2}\,ds . \end{aligned}$$

(5.15)

It follows from (5.14) and (5.15) that for $m > n\rightarrow \infty$,

$$ \bigl\Vert \partial _{t}v_{n,m}(\tau ,\tau -t_{n},\theta _{-\tau}\omega ) \bigr\Vert ^{2}+ \bigl\Vert v_{n,m}(\tau ,\tau -t_{n},\theta _{-\tau} \omega ) \bigr\Vert ^{2}_{H^{2}( \mathbb{R}^{n})}\rightarrow 0, $$

which together with (5.11) implies $\{\tilde{v}(\tau ,\tau -t_{n},\theta _{-\tau}\omega ,u^{(n)}_{0}), \tilde{v}_{t}(\tau ,\tau -t_{n},\theta _{-\tau}\omega ,u^{(n)}_{1,0}) \}^{\infty}_{n=1}$ is a Cauchy sequence in $H^{2}(\mathbb{R}^{n})\times L^{2}(\mathbb{R}^{n})$. This complete the proof. □

Lemma 5.4

Suppose (4.8) and (4.9) hold, then (5.1) has a unique $\mathcal{D}$-pullback random attractor $\mathcal{A}_{1}=\{\mathcal{A}_{1}(\tau ,\omega ):\tau \in \mathbb{R}, \omega \in \Omega \}\in \mathcal{D}$ in $H^{2}(\mathbb{R}^{a}n)\times L^{2}(\mathbb{R}^{n})$, which is actually a singleton; that is, $\mathcal{A}_{1}(\tau ,\omega )$ consisting of a single point for all $\tau \in \mathbb{R}$, $\omega \in \Omega$.

Proof

From Lemmas 5.2 and 5.3 by applying the abstract results in [19], we can obtain the existence and uniqueness of the $\mathcal{D}$-pullback random attractor $\mathcal{A}_{1}\in \mathcal{D}$ of (5.1) in $H^{2}(\mathbb{R}^{n})\times L^{2}(\mathbb{R}^{n})$ immediately.

Next, we prove $\mathcal{A}_{1}$ is a singleton. Suppose $\{t_{n}\}^{\infty}_{ n=1}$ 1 is a sequence of numbers such that $t_{n}\rightarrow \infty$ as $n\rightarrow \infty$. Given $\tau \in \mathbb{R}$, $\omega \in \Omega$, let $(z^{(n)}_{0},z^{(n)}_{1,0}), (y^{(n)}_{0},y^{(n)}_{1,0})\in \mathcal{A}_{1}(\tau -t_{n},\theta _{-t_{n}}\omega )$.

Similar to (5.13) we have

$$\begin{aligned} & \bigl\Vert \tilde{v}_{t} \bigl(\tau ,\tau -t_{n},\theta _{-\tau}\omega ,z^{(n)}_{1,0}\bigr)- \tilde{v}_{t} \bigl(\tau ,\tau -t_{n},\theta _{-\tau} \omega ,y^{(n)}_{1,0}\bigr) \bigr\Vert ^{2} \\ &\qquad {}+2 \bigl\Vert \Delta \tilde{v}\bigl(\tau ,\tau -t_{n},\theta _{-\tau}\omega ,z^{(n)}_{0}\bigr)- \Delta \tilde{v} \bigl(\tau ,\tau -t_{n},\theta _{-\tau}\omega ,y^{(n)}_{0}\bigr) \bigr\Vert ^{2} \\ &\qquad {}+ \nu \bigl\Vert \tilde{v}\bigl(\tau ,\tau -t_{n},\theta _{-\tau}\omega ,z^{(n)}_{0}\bigr)- \tilde{v}\bigl( \tau ,\tau -t_{n},\theta _{-\tau}\omega ,y^{(n)}_{0} \bigr) \bigr\Vert ^{2} \\ &\quad \leq e^{-\frac{1}{2}\varepsilon t_{n}}\bigl( \bigl\Vert z^{(n)}_{1,0}-y^{(n)}_{1,0} \bigr\Vert ^{2}+ \bigl\Vert z^{(n)}_{0}-y^{(n)}_{0} \bigr\Vert ^{2}+ \bigl\Vert \Delta z^{(n)}_{0}- \Delta z^{(n)}_{0} \bigr\Vert ^{2}\bigr) \\ &\quad \leq 2e^{-\frac{1}{2}\varepsilon t_{n}}\bigl( \bigl\Vert z^{(n)}_{1,0} \bigr\Vert ^{2}+ \bigl\Vert z^{(n)}_{0} \bigr\Vert ^{2}_{H^{2}(\mathbb{R}^{n})}+ \bigl\Vert y^{(n)}_{1,0} \bigr\Vert ^{2}+ \bigl\Vert y^{(n)}_{1,0} \bigr\Vert ^{2}_{H^{2}(\mathbb{R}^{n})}\bigr) \\ &\quad \leq 4e^{-\frac{1}{2}\varepsilon t_{n}} \bigl\Vert \mathcal{A}_{1}(\tau -t_{n}, \theta _{-t_{n}}\omega ) \bigr\Vert ^{2}_{H^{2}(\mathbb{R}^{n})\times L^{2}( \mathbb{R}^{n})}. \end{aligned}$$

(5.16)

Due to $\mathcal{A}_{1}\in \mathcal{D}$, we see that the right-hand side of (5.16) tends to zero as $n\rightarrow \infty$, and thus we obtain

$$\begin{aligned} &\lim_{n\rightarrow \infty}\bigl(\tilde{v}_{t} \bigl(\tau ,\tau -t_{n},\theta _{- \tau}\omega ,z^{(n)}_{1,0} \bigr)-\tilde{v}_{t} \bigl(\tau ,\tau -t_{n},\theta _{- \tau}\omega ,y^{(n)}_{1,0}\bigr)\bigr)=0 \quad \text{in } L^{2}\bigl(\mathbb{R}^{n}\bigr), \\ &\lim_{n\rightarrow \infty}\bigl(\tilde{v} \bigl(\tau ,\tau -t_{n}, \theta _{- \tau}\omega ,z^{(n)}_{0}\bigr)- \tilde{v}_{t} \bigl(\tau ,\tau -t_{n},\theta _{- \tau} \omega ,y^{(n)}_{0}\bigr)\bigr)=0 \quad \text{in } H^{2} \bigl(\mathbb{R}^{n}\bigr), \end{aligned}$$

which, together with the invariance of $\mathcal{A}_{1}$, shows that the $\mathcal{D}$-pullback random attractor $\mathcal{A}_{1}$ is a singleton. This complete the proof. □

To obtain the asymptotic compactness of the solutions of (5.2), we need the following Lemma.

Lemma 5.5

Let $u_{0}\in H^{2}(\mathbb{R}^{n})$, $u_{1,0}\in L^{2}(\mathbb{R}^{n})$, $\tau \in \mathbb{R}$, $\omega \in \Omega$ and $T>0$. If (3.2), (3.3), (3.6), (4.1), (4.2), and (4.5)–(4.8) hold, then the solution of (5.2) satisfies, for all $t\in [\tau ,\tau +T]$,

$$ \bigl\Vert A^{\frac{3}{4}}v(t,\tau ,\omega ) \bigr\Vert + \bigl\Vert A^{\frac{1}{4}}v_{t}(t,\tau , \omega ) \bigr\Vert \leq C, $$

where C is a positive number depending on τ, ω, T and R when $\|(u_{0},u_{1,0})\|_{H^{2}(\mathbb{R}^{n})\times L^{2}(\mathbb{R}^{n})} \leq R$.

Proof

This is an immediate consequence of Lemma 4.3. □

Lemma 5.6

Let (3.2), (3.3), (3.6), (4.1), (4.3), and (4.5)–(4.9) hold. Then, the cocycle Φ is $\mathcal{D}$-pullback asymptotically compact in $H^{2}(\mathbb{R}^{n})\times L^{2}(\mathbb{R}^{n})$; that is, the sequence $\{\Phi (t_{n},\tau -t_{n},\theta _{-t_{n}}\omega ,(u^{(n)}_{0},u^{(n)}_{1,0}) \}^{\infty}_{n=1}$ has a convergent subsequence in $H^{2}(\mathbb{R}^{n})\times L^{2}(\mathbb{R}^{n})$ for any $\tau \in \mathbb{R}$, $\omega \in \Omega$, $D\in \mathcal{D}$, $t_{n} \rightarrow \infty$, and $(u^{(n)}_{0},u^{(n)}_{1,0})\in D(\tau -t_{n},\theta _{-t_{n}}\omega )$.

Proof

Given $t\in \mathbb{R}^{+}$, $\tau \in \mathbb{R}$, $\omega \in \Omega$, and $(u_{0},u_{1,0}\in H^{2}(\mathbb{R}^{n})\times L^{2}(\mathbb{R}^{n})$, define

$$\begin{aligned} &\Phi _{1}\bigl(t,\tau ,\omega ,(u_{0},u_{1,0}) \bigr)=\bigl(\tilde{v}(t+\tau ,\tau , \theta _{-\tau}\omega ,u_{0}),\tilde{v}_{t}(t+\tau ,\tau ,\theta _{- \tau} \omega ,u_{1,0})\bigr), \\ &\Phi _{2}\bigl(t,\tau ,\omega ,(u_{0},u_{1,0}) \bigr)=\bigl(v(t+\tau ,\tau ,\theta _{- \tau}\omega ,u_{0}),v_{t}(t+ \tau ,\tau ,\theta _{-\tau}\omega ,u_{1,0})\bigr), \end{aligned}$$

where ṽ and v are the solutions of (5.1) and (5.2), respectively.

By (3.72) we have

$$ \Phi \bigl(t,\tau ,\omega ,(u_{0},u_{1,0})\bigr)=\Phi _{1}\bigl(t,\tau ,\omega ,(u_{0},u_{1,0})\bigr)+ \Phi _{2}\bigl(t,\tau ,\omega ,(u_{0},u_{1,0}) \bigr). $$

(5.17)

Let $B\in \mathcal{D}$ be the $\in \mathcal{D}$-pullback absorbing set of Φ given by (4.19). From Lemmas 4.2, 4.4, and 5.4 we see that for every $\delta >0$ there exists $t_{0}=t_{0}(\delta ,\tau ,\omega ,B)>0$ and $k_{0}=k_{0}(\delta ,\tau ,\omega )\geq 1$ such that for all $(u_{0},u_{1,0})\in B(\tau -t_{0},\theta _{-t_{0}}\omega )$,

$$ \bigl\Vert \Phi \bigl(t_{0},\tau -t_{0},\theta _{-t_{0}}\omega ,(u_{0},u_{1,0})\bigr)| _{ \tilde{\mathcal{O}}_{k_{0}}} \bigr\Vert _{H^{2}(\tilde{\mathcal{O}}_{k_{0}}) \times L^{2}(\tilde{\mathcal{O}}_{k_{0}})}< \delta , $$

(5.18)

with $\tilde{\mathcal{O}}_{k_{0}}=\{x\in \mathbb{R}^{n}:|x|>k_{0}\}$, and

$$ \Phi _{1}\bigl(t_{0},\tau -t_{0},\theta _{-t_{0}}\omega ,B(\tau -t_{0}, \theta _{-t_{0}}\omega ) \bigr) \quad \text{is covered by a ball of radius } \delta $$

(5.19)

in $H^{2}(\mathbb{R}^{n})\times L^{2}(\mathbb{R}^{n})$.

In addition, by Lemma 5.5 we know that for every $t\in \mathbb{R}^{+}$, $\tau \in \mathbb{R}$, $\omega \in \Omega$, and $k\in \mathbb{N}$,

$$ \Phi _{2}\bigl(t,\tau -t,\theta _{-t}\omega ,B(\tau -t, \theta _{-t}\omega )\bigr) \quad \text{is bounded in } H^{3}\bigl( \mathbb{R}^{n}\bigr)\times H^{1}\bigl( \mathbb{R}^{n} \bigr), $$

and thus for each $k\in \mathbb{N}$,

$$ \Phi _{2}\bigl(t,\tau -t,\theta _{-t}\omega ,B(\tau -t, \theta _{-t}\omega )\bigr)|_{ \mathcal{O}_{k}} \quad \text{is precompact } H^{2}(\mathcal{O}_{k}) \times L^{2}( \mathcal{O}_{k}), $$

(5.20)

with $\mathcal{O}_{k}=\{x\in \mathbb{R}^{n}:|x|< k\}$.

It follows from (5.17)–(5.20) that all conditions of Theorem 2.1 are satisfied, hence, Φ is $\mathcal{D}$-pullback asymptotically compact in $H^{2}(\mathbb{R}^{n})\times L^{2}(\mathbb{R}^{n})$. □

Since Lemma 4.2 implies a closed measurable $\mathcal{D}$-pullback absorbing set for Φ, and Φ is $\mathcal{D}$-pullback asymptotically compact in $H^{2}(\mathbb{R}^{n})\times L^{2}(\mathbb{R}^{n})$ from Lemma 5.6, we immediately obtain the following existence theorem by Theorem 2.2.

Theorem 5.1

Let (3.2), (3.3), (3.6), (4.1), (4.3), and (4.5)–(4.9) hold. Then, the cocycle Φ has a unique $\mathcal{D}$-pullback random attractor in $H^{2}(\mathbb{R}^{n})\times L^{2}(\mathbb{R}^{n})$.